Class Book A. \ '^ \ Ahl CoKii^ht^?. CflBauGwr BEPOsm .^ Handbook of Engineering Mathematics BY WALTER E. WYNNE, B. E. AND WILLIAM SPRARAGEN, B. E. lis ILLUSTRATIONS NEW YORK D. VAN NOSTRAND COMPANY 25 Park Place 1916 Copyright, 1916, BY D. VAN NOSTRAND COMPANY JAN -2 1917 Stanbopc jprcss F. H. GILSON COMPANY BOSTON, U.S.A. CI,A453449 AUTHORS' NOTE The authors have to express their thanks to Profes- sors Irving P. Church, G. A. Goodenough, and William A. Granville, who have kindly given permission for the use of special material, tables, and constants from their works, and to whom proper credit is given where such material appears. Thanks are also due to John D. Ball, of the Consulting Engineering Department of the Gen- eral Electric Company, for coefficients of hysteresis loss in iron. The authors are especially indebted to Professors Ernst J. Berg and John N. Vedder of Union College, and to Professor William D. Ennis of the Brooklyn Polytechnic Institute, for a critical reading of the manuscript and for valuable suggestions; also to Pro- fessors Charles F. F. Garis and Walter L. Upson of Union College for advice in connection with certain sections. August, 1916. Ill INTRODUCTION BY ERNST J. BERG, Sc. D. Professor of Electrical Engineering in Union College, Consulting Engineer of the General Electric Company, Schenectady, N. Y, In publishing this book the authors have endeavored to supply a handy means of reference to theoretical and apphed mathematics used in engineering, and while the first aim has been to make this a mathematical hand- book, the book is of greater value because it includes the imderlying engineering data and applications as well as the mathematical formulae. It is intended primarily for students in engineering schools and colleges, and should serve as a convenient reminder of things which are easily forgotten but are likely to be needed in their later work. In including differential equations, the authors have gone as far as seems necessary for students and en- gineers who have taken the ordinary undergraduate college course. The increasing need of mathematics in engineering should assure this book a broad field of usefulness, not only to students in technical schools and colleges but also to practising engineers. E. J. B. TABLE OF CONTENTS Page Algebra i Exponents i Binomial theorem 2 Progressions 2 Logarithms . , 3 Series 5 Complex imaginary quantities 7 Geometry 10 Plane figures 10 Solids 12 Plane Trigonometry 13 Niunerical values 14 Trigonometric formulae 14 Plane triangles 16 Spherical Trigonometry 16 Formulae 16 Application to navigation 17 Plane Analytic Geometry 19 Straight line 19 Rectangular and polar coordinates 20 Circle 21 Parabola 22 Ellipse 23 Hyperbola 24 Cycloid 25 Epicycloid 26 Hypocycloid 26 Catenary 27 Solid Analytic Geometry 27 Direction cosines 27 vii VIU TABLE OF CONTENTS Page Plane 29 Straight line 30 Calculus 30 Application of differential calculus 30 Formulae of differential calculus 33 Maxima and minima 34 Application of integral calculus 36 Curve tracing 40 Methods of integration 41 Table of integrals 46 Hyperbolic Functions 54 HyperboUc transformations 54 Hyperbolic formulae 55 Inverse hyperbolic functions 56 Differentials of hyperboHc functions 57 Differential Equations 58 Equations of the first order, and first degree 59 Equations of the first order, higher than the first degree 62 Linear equations with constant coefficients 63 Homogeneous linear equations 68 Exact differential equations 69 Equations of the second order and first degree 70 Theoretical Mechanics 72 Center of gravity 72 Moment of inertia 75 Motion of a body 78 Falling bodies 80 Projectiles 80 Impact .' 86 Forces 87 Friction 88 Inclined plane. 89 Mechanics of Materials 91 Direct stress , 92 Strength of materials 93 Torsion 94 Flexure of beams 95 TABLE OF CONTENTS IX Page Shear 97 Beam loadings 98 Columns 106 Center of gravity iii Moment of inertia of solids 115 Properties of standard sections 116 Hydraulics 119 Head and pressure 119 Center of pressure 119 Flow through apertures 120 Flow of water in pipes 121 Losses in pipes 122 Flow through open channels 127 Flow over weirs 129 Stresses in pipes and cylinders 130 Flow or Fluids 131 Flow through apertures 131 Flow through pipes 132 Electricity 134 Resistivity 134 Temperature coefficient 135 Copper wire tables 136 Motors and generators 139 Induced voltage 142 Inductance 143 Capacity 145 Alternating currents 147 Magnetism 154 Measurement 158 English weights and measures 158 English and metric conversion factors 160 Heat, electric, and mechanical equivalents 164 Pressure equivalents 165 Pressure and volimie corrections 166 Physical and Chemical Constants 168 Atomic weights 168 Weight and density of various substances 169 X TABLE OF CONTENTS Page Melting and boiling points 172 Specific heats 173 Coefficients of linear expansion of solids 176 Properties of saturated steam 177 Tables 185 Circumferences and areas of circles 185 Powers, roots, and reciprocals 187 Common logarithms 192 Natural logarithms 194 Trigonometric functions 197 Hyperbolic sines and cosines 213 Engineering Mathematics ALGEBRA Quadratic Equations ax^+ bx + c = X = ^ 2 a. The term b^ — 4: ac, called the discriminant, deter- mines the nature of the roots. If 6^ is greater than 4 ac, the roots are real. If W' is less than 4 ac, the roots are imaginary. And if 6^ = 4 ac, the roots are real and equal. Exponents 1 1 a^ = — a~^ = — {cry = a'^ri Special and Indeterminate Forms a« = 1 ^0°= cx), a> \ a-°° = -4: = — = 0, a>\ - = 00 — = 00 00 _ — =00 - = a a • 00, 77, — , 0^ 1°^, 00^, 00—00 are indeterminate. ' oo' For the evaluation of indeterminate forms, see page 38. 2 ENGINEERING MATHEMATICS Binomial Theorem /yt fyyt 1 J {x + yY = x^ + nx^'-^y H ^-y^ — - x^'-'^y^ . nin — V) (n— 2) ^ « , , + -^ ^ x'^-^y^ + • • • (\+xY=\+nx-{- ^ ' x^+— 3!~^ "*" ' * ' Arithmetical Progression An arithmetical progression is one whose terms increase or decrease by a common difference, a, a-\- dj a-\- 2d, a-\r ^ d, . . . the last term is L = a-{- {n — 1) d the sum of the terms is S = '^{a + L) ='^[2a+ in- \) d] a = first term n = number of terms d = common difference Geometrical Progression Quantities are in geometrical progression when each term is equal to the preceding term multipUed by a constant, a, ar, ar^, ar^, . . . the last term is L = ar""'^ the sum of the terms is _ a (f^ — 1) _ g (1 — r^) _ rL— a r — 1 1 — r r— 1 a = first term r = constant ratio n = number of terms ALGEBRA 3 The sum of an infinite number of terms in geometrical progression is S= ^ l-r in which the ratio r must be less than 1 if the series is to be convergent (see Infinite Series) . Logarithms The logarithm of any number to a given base is the power to which the base must be raised in order to produce the given number, thus: if x"^ = y, then m = logxy, that is, m is the logarithm of y to the base x. The following relations hold for any base: log ab = log a + log b log - = log a - log b log a^ = n log a log- = - loga The base of the common system of logarithms is 10. The base of the natural system of logarithms (also called Naperian or hyperboHc logarithms) is 6 = 2.7182818284 .... A logarithm may be transformed from any given base to any other desired base by the relation: logaA^ logaft To transform a logarithm from base 10 to base e, 4 ENGINEERING MATHEMATICS multiply by 2.302585 . . . (where 2.302585 . . . is the logarithm of 10 to the base e): logea = 2.302585 logio^ To transform a logarithm from base e to base lo, divide by 2.302585: logioa = ^ 3Q2535 ^^S^^ = 0.434294 logefl Special forms : log 1 =0 (to any base) loga a = I log^e = 1 log = —QO log 00 = 00 Cubic and Higher Degree Equations The approximate values of the real roots of an alge- braic equation containing only one variable may be found graphically. For instance, let it be required to solve the equation 0(? + Ax — B = 0, This may be written sls o(? = — Ax+ Bj or as two simultaneous equations y = o(? and y = —Ax + B, The graph of each of these equations being plotted, the abscissas of their points of intersection give the real roots of the cubic. The curve y = o(^ should be plotted on cross-section paper by the aid of a table of cubes. The curve y = —Ax + B is the equation of a straight line, and is therefore determined by plotting two points. Algebraic equations of any degree may be solved by Newton's method of approximation; see page 39. Transcendental Equations The graphic method given under Cubic and Higher Degree Equations is also applicable to many trans- ALGEBRA S cendental equations. Thus, the equation Ax — sin :^ = may be solved by plotting the two simultaneous equations y = Ax and y = sin x. The curve y = sinx is readily plotted with the aid of a table of sines, while the other curve y = Ax is a straight Une passing through the origin. Infinite Series An infinite series is one containing an unlimited number of terms. Such a series is convergent if the sum of its terms is a finite quantity. It is divergent when the sum of its terms does not approach a finite limit. Comparison Test. A series is converging if each term in it is equal to or less than the corresponding term of a known converging series. Converging series for comparison: a-\- ar -\- ar^ -{• ar^ + • • • -\-ar^''^-\- • • • [^<1^ ■+1 + 1+...+ • • 1-2 ' 2-3 ' 3.4 ' ' w(«+ 1) ' l + ^ + f,+ •••+;^+ ••• [/»!] A series is diverging if each term in it is equal to or greater than the corresponding term of a known di- verging series. Diverging series for comparison: a+ ar + ar^ + ar^+ • • • + ar"""^ -|- . . . [r = 1] 1+1+1 + 1 + 1+ . . . Ill 1 l + - + i + i+ •••+-+... 6 ENGINEERING MATHEMATICS Ratio Test. If, as the number of terms approaches infinity as its Umit, the ratio of the (n + l)th term to the nth term approaches some finite limit (a), the series is convergent if (a) is less than 1, divergent if (a) is greater than 1, and indeterminate by this method it (a) = 1. Oscillating Series. A series whose terms are alter- nately positive and negative is convergent if each term is numerically less than the preceding term. Standard Series /y^ /yO /y«4 e^ = l + . + ^, + | + |;+... /yi /yO /yft e-=l-^ + 2j-3j + jj , /y^ /%/yO /yi 1+1+1+1+ = 2.7182818 . . . _ limit / , ly /yi /yiO /y«4 log(l + x)=x-| + J-|-+ . . . [lSiC>-l] /y»« /yO /yi. log(l-x) = -x-|-|-J- ... [1>«S-1] /yO /y^ /yl /yiV) ALGEBRA X^ . 0(^ X' T cosx - 1 2!~'~4! 6! "*" 8! 11% ' + ^^ A. 2835 cot a; = - X X 3 45 2x' X' 945 4725 TT COS ^ X = - — sin~^ X 2 tan*"^ x = x — — + -F- sinhx = X + jj + || + yj' + ^ + COSh^=l+2;+4;+^+gj + X' 5x^"^ X' • • [x2<7r2] . fl>x>-l] [1>X>-1] 1 =J Complex Imaginary Quantities The imaginary unit = V- In representing complex Y imaginary quantities, it is usual to represent real quantities in the direction of the horizontal or X- axis, and imaginaries in the direction of the verti- cal or F-axis. Multipli- 8 ENGINEERING MATHEMATICS cation by the imaginary unit, j, revolves a quantity through 90 degrees, in counter-clockwise direction. A complex number is the sum of a real and an imaginary, thus: A =a+jb ^a+V^b is a complex number. A complex number may be written in any of the following identical forms: ^ = a+jb = r (cos0 + y sind) = re^ [6 in radians] a = r cos 6, b = r sin 6. in which The magnitude of the complex number, a + jb, is r = Va" + b^ Addition and Subtraction of complex quantities: To add two complex quantities, combine the real parts, and then the imaginaries, thus: (a +jb) + {c +jd) = {a+c) +j {b + d) In the same way, to subtract two complex quantities: {a+jb) - {c +jd) = {a-c)+j{b- d) Multiplication of complex quantities: To find the product of two complex numbers, mul- tiply out as in ordinary algebra, remembering that p = — 1, thus: {a +jb) (c+jd) = {ac - bd) +j (ad + be) Division of complex quantities: To divide two complex quantities, rationalize the denominator as follows: a+jb _ a+jb ^ c — jd _ (ac + bd) + j {bt — ad) c + jd c + jd c — jd c^ + d^ ALGEBRA 9 Exponential Transformations smax = cos ax = 2j ^j'ax I ^—jax (e is the base of the hyperbolic logarithms; j equals De Moivre's Theorem: (cos +y sin Oy = cos nd+jsinnd Permutations and Combinations The number of permutations of n different things taken r at a time is nl Pr =n{n- I) . . . {n-r+ I) = {n — r)\ For n different things taken all at a time, the number of permutations is Pn = n{n-l) . . . (2) (1) =nl The number of permutations of n things taken all at a time, ni being alike, n-z alike, ns alike, etc., is P= — ... fill fhl fisl The number of combinations of n things taken r at a time is _n{n— I) . . . (n — r + 1) __ n\ *" r\ r\ {n^ r)\ For n things taken 1, 2, 3, . . . /^ at a time, the total number of combinations is C = 2^ - 1 lO ENGINEERING MATHEMATICS GEOMETRY Plane Figures Right Triangle h = Vc2 — a2 area _ 1 a& Any Triangle _ 1 area = ^bh area = Vs{s a) {s - h) {s c) Parallelogram area = ah Trapezoid area = \h{a + h) Regixlar Polygon area = J ahn n = number of sides GEOMETRY II Circle circumference = 2 Trr = wd area = irr^ (P = -4 Sector of Circle arc = I = irr 180° area = 4 r/ = wr^ 360° Segment of Circle chord = c = 2 V 2 ^r - h^ area = | r/ — | c (r — /f) Parabola length of arc = 3-7 [v c (1 +c) + 2.0326 logio (V^+ VI+"c)] in which Ellipse circumference = 64-3 Tr(a + b) fb-_aY [b-^aj 64 -K^:y (close approximation) area = irab 12 ENGINEERING MATHEMATICS SoUds Right Prism lateral surface = perimeter of base X h volume = area of base X h < > Pyramid lateral area = J perimeter of base X / h volume = area of base X -= Frustum of Pjn'amid lateral surface = \l(JP -\- p) P = perimeter of lower base p — perimeter of upper base volume = lh[A + a-{- V Aa] A = area of lower base ^ a = area of upper base Right Circular Cylinder lateral surface = 2 irrh r = radius of base volume = irr^ Right Circular Cone lateral surface = irrl r = radius of base PLANE TRIGONOMETRY 13 Frustum of Right Circular Cone lateral surface = tt/ (i? + r) R = radius of lower base r = radius of upper base volume = iirh [R^ + Rr + r^] Sphere surface = ^tirr^ volume = ^irr^ Segment of Sphere volume of segment L to / IP^X PLANE TRIGONOMETRY Right Triangle sin^ = tan^ = cos^ = cot ^ = - a sec ^ =7 sin A = cos cosec^ = a cos 14 ENGINEERING MATHEMATICS tan A =- cot I - — ^ cot^ = tan (^-a)^- ^^^(|+ a A = secf^- ^j= - sec\^+ Aj sec A = cosec cosec sin {^ A) = — sin A tan(— A) = — taxiA sec (— A) = sec A cos (— A) = cos A cot {— A) = — cot A cosec (— A) = — cosec i4 NUMERICAL VALUES Angle . . 0° 30° 45° 60° 90° sin 1 2 V2 2 Vs 2 1 cos 1 V3 2 V2 2 1 2 tan V3 3 1 Vs 00 cot 00 Vs 1 Vs 3 Trigonometric Formulae tanx = sec:i: = tan jc = sin:;i[; cos:r 1 cosx 1 cot:x: cotx = cosx sinx 1 cosec X = cotx = smo; 1 tan jc PLANE TRIGONOMETRY 1 5 sin^ X + cos^ X = I sec^ X = 1 + tan^ x cosec^ X = 1 + cot^ X sin {x + y) = sin x cos y + cos x sin 3; cos (x + y) = cos :^ cos y — sin x sin y , , V tan X + tan y tan (x + y) = z 7 r-^ 1 — tan X tan y ^ , , X cot X cot 3; — 1 cot (^ + }') = — I — r^S: — cot X + cot 3^ sin (x — 3^) = sin X cos y — cos x sin y cos (x — 3^) = cos X cos 3; + sin X sin y tan X — tan 3; tan (x — y) = cot (x — y) = 1 + tan X tan y cot X cot 3;+ 1 cot y — cot X sin 2 X = 2 sin X cos x cos 2 X = cos^ X ~ sin^ x 2 tanx tan2x = cot 2 X = 1 — tan^ X cot^ X — 1 2 cotx sinf .-si'- 1 . / - ~ cos X + cosx cos tan|x = 2 1 — cos X sinx sin X + sin y = 2 sin i^ (x + 3;) cos | (x — y) sin X — sin y = 2 cos | (x + y) sin J (x — y) cos X + cos y = 2 cos I (x + y) cos hi^-^ y) cos X — cos y = — 2 sin I (x + y) sin | (x — y) i6 ENGINEERING MATHEMATICS a Solution of Any Plane Triangle b c sin A sin B sin C a^ ^b^^ c^ — 2bc cos A a-b ^ tan ^ {A - B) a + b~ tan^iA + B) a sin C tSLYiA == b — a cos C SPHERICAL TRIGONOMETRY Right Spherical Triangles cos c = cosa cos b sin a = sin c sin A sin b = sin c sin 5 cos A = cos a sin 5 cos B = cos i sin ^ cos A = tan b cot c cos B = tan a cot c sin & = tan a cot A sin a = tan b cot B cos (; = cot A cot 5 Oblique Spherical Triangles i sma sin 6 smc sin ^ sin 5 sin C cos a = cos&cosc + sin6sin(;cos^ cos ^ = sin5sinCcosa— cos5co^C cot a sin 6 = cot-4 sin C+cos C cos 6 ^ s = ^{a + b + c) SPHERICAL TRIGONOMETRY 17 sin (i) = J sin(5-^,)sin(.- g \2 / V sin b sm c M \ 4 /sin ^ sin (5 — a) cos h^ = V ^^T^ \2/ V sin 6 sine tan (-] = 4 AiJ^ (^ - b) sin (5 - c) \ 2 / V sin 5 sin (^ — a) . /a\ . / cos S cos (5 — ^) ^^" (2 j = V - sin ^ sin C it)-^' ^ _ ^ , cos (5 — B) cos (5 — C) sin ^ sin C ©=v/ , - . . , COS 5 COS (S — A) tan ' » — * ' cos (5-^) cos (5- C) ^ 1 , - . sin n^ - ^) ^ 1 tan^(a-&)=^.^,^^_^^^tan|c 1 / . 7 \ cos h (A — B) . tan^(a+5)== ^^^,^^^^) tan^. tan H^ - ^) = 'l^'I^^'lS c<^t I C sm I (a + 6) tani (A + B)= ^^4fe|! cotK cos I (a + 6) 1 sinH^ + B) tSLiij (a - b) tan ^ c = ; — J— r-j ^T sm^ (A — B) Application of Spherical Trigonometry to Navigation To find the shortest distance between two points on the earth's surface and the bearing of each from the other, the latitude and longitude of each being given. (From W. A. Granville's '^ Plane and Spherical Trigo- nometry.") l3 ENGINEERING MATHEMATICS (i) Subtract the latitude of each place algebraically from 90°, taking North latitudes as positive and South latitudes as negative. The results will be the two sides of a spherical triangle. (2) Find the difference of longitude of the two places by subtracting the lesser longitude from the greater if both are East or both are West; but adding the two if one is East and the other West. This gives the included angle of the triangle. If the difference of longitude found is greater than 180°, then subtract it from 360° and use the remainder as the included angle. (3) Solving the triangle by the formulae for tan | {A — B)j tan ^ {A + B), and tan | c, the third side gives the shortest distance between the two points in degrees of arc, and the angles give the bearings. The number of minutes in the arc will be the distance between the places in nautical miles. Illustration. Find the shortest distance along the earth's surface between Boston (latitude 42°21'N., Nojth Pole South Pole longitude 71° 4' W.) and Capetown (latitude 33^56' S., longitude 18° 26' E.) and the bearing of each city from the other. (1) a = 90°- 42° 21' = 47° 39' J = 90° - (-33° 56') = 123° 56' PLANE ANALYTIC GEOMETRY 19 (2) C = 71° 4' + 18° 26' = 89° 30' = difference in longitude. (3) Solving the triangle as explained above, we get c = 68° 14' = 68.23° = 4094 nautical miles. A = 52° 43' = bearing of Boston from Cape- town. B = 116° 43' = bearing of Capetown from Boston. PLANE ANALYTIC GEOMETRY The Straight Line I. The slope equation: y = MX + b m = slope = tan^ b = intercept on F-axis n. The intercept equa- tion: where a and b are the inter- cepts on the X and F-axes. III. Line through the points (x',/) and {x'\y"): y — y' X — x' y" - y' X X (x',y') (x".y") •X 20 ENGINEERING MATHEMATICS Y IV. Line through the point {%', y), with slope m: y ^ y =^ m{x — x') V. Distance from the point {x',y') to the line Ax + By + C = Q: Ax' + 5/ + C d = ± V^2 4. ^2 VI. Distance between the points {x', y') and rf = V(x' - x'y +{y' - y"y Transformation from Rectangular to Polar Coordinates X = r cosd y = rsind r = radius vector = Vx^ + y^ = polar angle = tan~- r Vx^ + / X PLANE ANALYTIC GEOMETRY 21 The Circle I. Circle of radius r with cen- ter at origin: ^ + y^ = y^ II. Circle of radius r with its center at the point (a, b) : *— a III. Tangent at the point (a, 6) of the circle o^ + y2 _ ^2 jg ax-\-hy ^ r^ IV. Slope equation of the tangent to the circle iX^ -f y2 _ ^2 Jg y = mx zh r v w^ + 1 V. Polar equation of circle of radius a passing through the origin, and having its center on the X-axis: r = 2 a cos 22 ENGINEERING MATHEMATICS VI. Polar equation of circle of radius a passing through the origin, and having its center on the F-axis: r = 2 a sin Parabola Definition. The parabola is the curve generated by a point moving so as to remain always equidistant from a given fixed point and a given fixed line. The fixed point is called the focus ; the fixed line is called the directrix. I. Parabola with its v axis along the X-axis and vertex at origin: y^ = 4:ax where a is the distance from the origin to the focus. II. Parabola having its axis along the F-axis and vertex at origin: x^ = 4:ay where a is the distance from the origin to the focus. ~ III. General equation of a parabola with axis parallel to the Z-axis: X = ay^+ by + c PLANE ANALYTIC GEOMETRY the vertex is at the point 23 / P-4ac _ _6_\ V 4a ' 2 a) IV. General equation of a parabola with axis par- allel to the F-axis: y = ax^+ bx + c the vertex is at the point / _&_ b^-4:ac \ [ 2a' 4a / V. Slope equation of the tangent to the parabola y^ = 4 ax is y = mx H — VI. Slope equation of the tangent to the parabola ^2 = 4 ay is y = mx — arr? Ellipse Definition. The ellipse is the curve generated by a point moving so that the sum of its distances from two fixed points is always constant. The fixed points are called the foci. I. Equation of el- lipse with center at origin: i ^^ _i_ 3^^ _ 1 j where a and h are one- half the major and minor axes. 24 ENGINEERING MATHEMATICS II. Slope equation of the tangent to the ellipse -^ + 7? = 1 IS a^ 62 Hyperbola Definition. The hyperbola is the curve generated by a point moving so that the difference of its distances from two fixed points is always constant. I. Equation of hyperbola with center at origin: n. Equation of conjugate hyperbola: x^ y^ a^ ¥ = -1 ><^ PLANE ANALYTIC GEOMETRY 25 III. Equations of asymptotes of the hyperbola -:; — 7^ = 1 are b y = -X a b — -X a IV. Slope equation of the tangent to the hyperbola x^ f . "2 -^ = 1 IS y = mx dz ^c^yr^ — b^ V. Slope equation of the tangent to the conjugate hyperbola ^ - ^ = - 1 is y = mx zb V&2 — a%2 Cycloid Definition. The cycloid is the curve generated by a point on the circumference of a circle as the circle rolls along a straight line. a: = a (0 ~ sin 0) y = a (1 — COS0) ri^-V2 X = a vers~" ' — v z ay a where a is the radius of the rolling circle. 26 ENGINEERING MATHEMATICS Epicycloid Definition. The epicycloid is the curve generated by a fixed point on the circumference of a circle which rolls externally on the circumference of a fixed circle. X =^ {a+ b) COS0 — b cos ( — 7 — dj y = {a+ b) smd - bsm(^^ e) where a is the radius of the fixed circle, and b the radius of the rolling circle. Hypocycloid Definition. The hypocycloid is the curve generated by a point on a circle which rolls internally along the circumference of a fixed circle. X = (a— b) cosd + b cos f — t~^) SOLID ANALYTIC GEOMETRY 27 y = (a — b) sind — b sin ( — ; — 6 (^'') where a is the radius of the fixed circle and b the radius of the rolling circle. The Catenary The catenary is the curve which a heavy cord or perfectly flexible chain of uniform density forms, due to its own weight, when freely suspended between two. points. a y = 2 v^'' + ^ ) = a cosh SOLID ANALYTIC GEOMETRY The direction cosines of a line in space passing through the origin are the cosines of the angles which the line makes with the rectangular coordinate axes. The direction cosines of any line in space are the direction cosines of a line parallel to it and passing through the origin. 28 ENGINEERING MATHEMATICS I. Distance from the point {x, y, z) to the origin; r = Vx^ + / + ^2 z II. The direction cosines of the line from the point (x, y^z) to the origin are: (x,y,z) cos a = - = i)C r Vx2 + / + >^^-i-s^ COSjS _3^_ r Vx2 + / + z2 cos 7 = - = r V:r2 +3,2+22 III. The sum of the squares of the direction cosines of a line is equal to 1, cos2 a + cos2 /3 + cos2 7=1 SOLID ANALYTIC GEOMETRY 29 IV. Distance between the points {x, y, z) and d = V{x- x'Y +{y- y'Y +{z- z'f V. Direction cosines of a line joining the points (x, y, z) and {x\ y\ z')\ X — X cos a = COSjS = cos 7 = «/v »a/ d V(x - xy +{y- yy +(z- zy y-y _ y-y d V(x - x'f +{y- yy + (2 - zy Z— Z^ 4^ Z— Z^ d V{x - xy +{y- yy +(z- zy VI. The angle between two Unas in terms of their direction cosines: cos ^ = cos a cos a + cos ^ cos jS' + cos 7 cos 7' Vll. Intercept equa- tion of a plane: a b c where a, &, and c are the intercepts of the plane on the Z, Y , smd Z axes. VIII. General equation of a plane : Ax + By + Cz+D = IX. Distance from the point (x\ y, z') to the plane Ax-\-By^Cz^B = ^\ Ax' + By ^Cz' ^D d = ± Va^ + 52 + C2 30 ENGINEERING MATHEMATICS X. Straight line through the two points {x'\ y'^ s'O and {x\ y\ z') : X — x' = y. y" -y -y' = z — z' x" - ■x' z"- -z' XI. Straight line through the point {%', y', z')» and making the angles a, jS, and 7 with the coordinate axes: X— x' _y — y' _z— z^ cos a cos /5 cos 7 XII. General equation of a straight line is given by the equations of two intersecting planes: A'x+ B'y+ Cz+ D' = A''x + B"y + C'z + D" = Q CALCULUS Application of Differential Calculus The following list includes some of the principal formulae necessary for the solution of geometrical and physical problems, relating to any curve y = f (x). Rectangular Coordinates : dv Slope of the tangent at the point {x, y) = -^ dx Slope of the normal = — j- Equation of the tangent at the point {xo, y©), Xo and yo being the coordinates of the given point, is dyo r V Equation of the normal at (xo, y^ is CALCULUS 31 dx The intercept of the tangent on the Z-axis hk — y — The intercept of the tangent on the F-axis \sy — x-~ ax The intercept of the normal on the Z-axis \^x-\- y-^ dx The intercept of the normal on the F-axis is 3; + ^ t- Length of the tangent from its point of contact with the curve to the Z-axis is M^ Length of the tangent from its point of contact with the curve to the F-axis is V'+(D* Length of the normal from its point of contact with the curve to the X-axis is \/'+(i)' Length of the normal from its point of contact with the curve to the F-axis is V'+(|J dx Length of the sub tangent = y-j- dy Length of the subnormal = J-r- Differential length of the arc = ds = V(dxy + (dy)^ 32 ENGINEERING MATHEMATICS Radius of curvature = h( dySff dx) \ d^ Curvature is the reciprocal of radius of curvature. Length of the perpendicular from the origin on the tangent (to the curve) is dy Polar Coordinates : tan \l^ = r -r-, where ^ is the angle between the radius vector and that part of the tangent to the curve at {r, 6) drawn back toward the initial line. Length of polar sub tangent = ^^ ir dr Length of polar subnormal = -jz uu Differential length of arc = ds = V(dry + r^ {dSf Length of the perpendicular from the pole on the tangent = p = r^-ri also, 1 = 1 1 1/^V A "^ CALCULUS 33 Formulae of Differential Calculus d iau) = adu d{u + v) = du-{- dv d (uv) = vdu + udv- J /^\ _vdu — udv d (x"^) = nx"^'^ dx d (x^) = yx^-^ dx + x^ \oge x dy d (e^) = e^dx d {a"") = a^ loge a du d (loge x) = -dx X d (sin x) = cos x dx d (cos x) = — sin x dx d (tan x) = sec^ x dx d (cot x) = — cosec^ x dx d (sec x) = sec x tan x dx d (cosec x) = — cosec x cot x dx dx d (sin~^ x) d (cos""^ x) == — d (tan~^ x) = d (cot~^ x) = d{sec~^x) = Vl-x^ dx Vl-x' dx l + x^ dx 1 + x^ dx x \^x^ — 1 dx d (cosec"^ x) = — / . ^ Vr2 _ 1 34 ENGINEERING MATHEMATICS Maxima and Minima The maximum or minimum values of a given func- tion y = / (x) are obtained as follows: (1) Find the first derivative -j- and equate it to zero. (2) Solve the resulting equation for values of x. (3) In order to determine whether these values of X make y maximum or minimum, obtain the second derivative -7^ of the given function. d^y (4) Substitute separately in the expression for ~~^ each of the values of x found above. Values of x that d^y , , make -7^ positive correspond to minimum values of d^y the function, and values of x that make -7^2 negative correspond to maximum values of the function. (5) Substituting these values of x in the given function y = f {x)y we obtain the maximum or minimum values of y. Illustrative Example. Find the values of x which will make the function y = 6x-\-3 x^— 4:X?sl maxi- mum or a minimum, and find the corresponding values of the function y. (1) The first derivative of 3/ is ^ = 6+6x-12x2 (2) The values of x which make y maximum or dy minimum will make ~f- = Oj therefore 6+ 6x— 12x^ = 0, or x^ — |x = | CALCULUS 35 solving, x = Jzbf = +l or — | Hence, the maximum or minimum values of y must occur when x = 1 or — |. (3) To determine whether these values are maxima or minima, we obtain the second derivative of y\ thus: (4) When x = 1, -j^ = — 18, which corresponds to a maximum value of y. When X = — i, -7^ = +18, which corresponds to a minimum value of y, (5) Substituting these values of x in the given func- tion, we have when x = l, ;y = 6+3 — 4 = 5, a maximum when X = — |, y = — 3 + f + | = — |, a minimum Taylor's and Maclaurin's Series Taylor's Series: where / denotes the function, /' the first derivative, f the second derivative, etc. Maclaurin's Series: /(x) =/(o)4-/(o)f +r (o)|J+r(o)^+ . . . where / (0) denotes the value of the function when is substituted for x, j' (0) the value of the first derivative when is substituted for x, etc. 36 ENGINEERING MATHEMATICS APPLICATION OF INTEGRAL CALCULUS Lengths of Curves Rectangular Coordinates : y 1 +[-f-] dx From the equation of the given curve, find y in terniL of x: then differentiate in order to obtain -r-, and ax- substitute its value in the formula. The lower limit a is the initial value of x, and the upper limit b the final value of X. Or, similarly, by solving for x in terms of y, and dx obtaining -^, the length of the curve is given by the formula 'J:mw^' where c and d are the initial and final values of y. Polar Coordinates : length of curve = 5=1 V/l + r^ ( — j dr where a and b are the limiting values of r. Or, length of curve = 5=1 V/ r^ ~^ \^) ^ where 6^ and 6^^ are the limiting values of 6. CALCULUS 37 Plane Areas Rectangular Coordinates : The area included between a curve, the Z-axis, and the vertical Hnes x = a and x = b is = A = I ydx tJ a area The value of y in terms of x is found from the given equation and substituted in the formula. The initial value of X is a, and the final value b. Similarly, the area included between a curve, the F- axis, and the horizontal lines y = c and 3/ = J is area = A = j xdy where c and d are the limits of y. Polar Coordinates : The area included between a given curve and two given radii is re" area = A = ^ I r^ dd where B" and B^ are the limiting values of B. Areas of Surfaces of Revolution For revolution about the X-axis, area A.2.£y^l + {fjd, where the value of (-1^) is found from the given equa- tion. The initial value of x is a, and the final value b. 38 ENGINEERING MATHEMATICS For revolution about the F-axis, area = A = lir \ ^\ ^-\-\-r\ ^V where c and d are the limiting values of y. Volumes of Solids of Revolution Rectangular Coordinates : volume = Fx = TT I y^ dx is the formula for the volume generated by revolving the given curve about the X-axis. The limiting values of X are a and b. Similarly, the volume generated by revolving the plane figure about the F-axis equals Vy = T I x^ dy where c and d are the initial and final values of y. Polar Coordinates : ' When the plane figure is revolved about the Z-axis, the volume generated is Va: = 2T C ir'^sinddddr For revolution about the F-axis, the volume generated is r^ cos 6 do dr • '\ff' INDETERMINATE FORMS f (x) If the fraction v. \ \ gives rise to the indeterminate form TT or — , when x approaches a as a limit, the in- CALCULUS 39 determinate form may be replaced by a new fraction, f (x) •L/ / X , the numerator of which is equal to the deriva- F {x) tive of the given numerator, and the new denominator is equal to the derivative of the given denominator. The value of this new fraction, as x approaches a, is the limiting value of the given fraction. If this again becomes indeterminate, it may be necessary to repeat the process several times. Example. Find the limiting value, when x = l, ot the fraction x^ + x-2 x^- I f(x) x^ + x-2 ^ t^T^ = ^ z — = ^, when x = 1 F {x) x^ — 1 fix) 2x+l 3 , ^ F {x) 2x 2 Hence, the required limiting value is f . SOLUTION OF EQUATIONS Algebraic equations may be solved by Newton's method of approximation. Thus, let it be required to solve an equation of the form Ax^ + Bx^ + Cx = D. Find, by trial, a number, r, nearly equal to the root sought, and letr -\- h denote the exact value of the root, where his sl small quantity the value of which must be determined. Substituting r + ^ for x in the given equation and neglecting all powers of /j higher than the first, we have, approximately, J Ar^ + Br^ + Cr - D -3Ar^-2Br-C It will be observed that the numerator of the above 40 ENCxINEERING MATHEMATICS fraction is the first member of the given equation after D has been transposed and x changed to r, and the denominator is the first derivative of the numerator with its sign reversed. The correction h added, with its proper sign, to the assumed root r, gives a closer approximation to the value of x. Repeat the opera- tion with the corrected value of r, and a second correc- tion will be obtained which will give a nearer value of the root; two corrections generally give sufficient accuracy. Illustration. Find a root of the equation The value of /? is r^ -f- 2 /^ + 3 r - 50 h = -3r^- 4:r By trial, we find that x is nearly equal to 3. On sub- stituting 3 for r, we have 2 /? = — — = — 0.1, approximately Hence, x = 2.9, nearly. If we substitute this new value of r, the new value of h equals +0.00228. Hence x = 2.90228. If we repeat the operation with this last value of r, the value of h is then found to be +0.0000034. Hence x = 2.9022834. CURVE TRACING The usual method of tracing curves consists in assigning a series of different values to one of the variables, and calculating the corresponding series of values of the other, thus determining a definite number of points on the curve. By drawing a curve through CALCULUS 41 these points, we obtain a graphical representation of the given equation. The general form and peculiarities of the curve can be easily determined and sketched by the following steps: (1) If possible, solve the equation of the given curve for one of its variables, y for example. If the equation then contains only even powers of x, it is symmetrical with the F-axis. Or if, when solved for x, it contains only even powers of y, it is symmetrical with the X-axis. (2) Find the points in which the curve cuts the axes by solving the equation of the given curve in turn with the equations x = and y = 0. (3) Find the values of x^ if any, which make y infinite; similarly, test for infinite values of :^. (4) Find the value of the first derivative ~-\ and ax thence deduce the maximum and minimum points of the curve. In tracing polar curves, write the equation, if possible, in the form r = f {d)\ and give d such values as make r easily found, as for example, 0, i tt, tt, | tt, etc. dr Putting 1^ = 0, we find the values of Q for which r is a maximum or minimimi. METHODS OF INTEGRATION (By parts, substitution, etc.) When the numerator of a fraction contains a variable to an equal or a higher power than the denominator, the fraction must be reduced to a mixed quantity (by 42 - ENGINEERING MATHEMATICS actually dividing the denominator into the numerator) before it can be integrated. If an expression cannot be integrated by the formulae given in the table of integrals, one of the following methods may be used to obtain a solution. Partial Fractions A fraction may be resolved into partial fractions, which can be integrated separately. Example. To integrate dx {x + a) (x + b) Let 1 ^ A B {x + a) ix +b)~ {x+a)'^ {x+b) where we must determine A and B. Clearing of fractions, \ = A {x+ b) + B {x+ a) =^ {A + B) x^- {bA + aB) The coefficients of like powers of x on both sides of the equation are equal; therefore, A + B = bA + aB ^\ whence A = f and B = 7 — a a — and J (x+a){x+b)'^''~J (x+a)'^^+J (x+ft)"^^ These forms are now integrable by the table of integrals, the result being /I 1 1 / , w , ,sd x = 7 log(x+a)H Aog(x+b)+C {x+a){x+b) b—a a — b °^ where C is the constant of integration. CALCULUS 43 Integration by Parts To integrate by parts, apply the formula I udv = uv — j vdu The method of integration by parts is most effective in dealing with the integration of products, involving logarithms, and trigonometric and inverse circular functions. Generally, the most complicated quantity which can be integrated directly by one of the fundamental for- mulae (see Table of Integrals, page 46) is equated, with the differential, to dv, and the remaining part is equated to u. Example. To find X log (x) dx I' Let u = log X and dv — xdx dx i oc^ then du ^ — v = I xdx = -^ X . J 2 Substituting in the formula \ udv = uv— \ vdu we have /-v" ioc^ dx X log (x) dx = \og{x) . 2" "" J 2" ^ = |-log(x)-|+C Integration by Substitution I. Differentials containing fractional powers of x may be integrated by the substitution X = 2" 44 ENGINEERING MATHEMATICS where n is the least common denominator of the frac- tional exponents of x. II. Expressions involving only fractional powers of {a + bx) may be rationalized by the substitution {a + bx) = z^ where n is the least common denominator of the frac- tional exponents of (a -\- bx). III. To integrate expressions containing V :x:^ + ax + &, use the substitution V x^ -\-ax-\'b = z — x IV. Expressions containing V— ^^ + ax + 6 may be rationalized by the substitution v— x^+ ax+ b = {x — 6) z where (x — 6) is sl factor of (— x^ + ax + b). V. A differential containing sinx and cosx can be transformed by means of the substitution ^ X tan - = s from which 2 z ^ 1-^ ^ 2dz sm X = T—. — ^ cos X = T— i — o dx = :i— ; — ^ 1 + 2^ 1 + r 1 + 2^ VI. A very useful substitution is 1 X = - 2 VII. Differentials involving Va^ — x^ may be ra- tionalized by the substitution X = a sin ^ VIII. Differentials involving Va^ + x^ may be ra- tionalized by the substitution X = a tan d CALCULUS 45 IX. Diflerentials involving Vx^ — c? may be ra- tionalized by the substitution X = a sec ^ Reduction Formulae The purpose of the following reduction formulae is to simplify an integral of the form / / x"^ (a + bx^ydx - , ^ , , , X ,, {np -\- m+ l)b (m — n-\- V) a C , , , . , {np + w + 1) 6 J This formula enables us to lower the exponent of x by n, without affecting the exponent of (a + bx^). Method fails when {np + w + 1) = 0. II. fx-^ {a + bx-y dx = ^7""' ^^ + ^"^^^"^ J {np + m+ I) + -' . . ,x / x'^{a+ bx'^y-^ dx {np -\- m+ i) J By this formula, the exponent of {a + bx^) is lowered by 1, without affecting the exponent of x. Method fails when {np + w + 1) = 0. III. / x^ (a + bx'^y dx = r — hrv^ — J {m+ l)a {m+ 1) a J By this formula, the exponent of x is increased by n^ without affecting the exponent of {a + bx""). Method fails when m = —1. 46 ENGINEERING MATHEMATICS IV. A;- (a + bx-y dx = _ ^"^'(^+M^+' J n{p+ l)a , (np + n+ m+ 1) r , . , ^ ^. ^ n{p+ l)a J This formula enables us to increase the exponent of (a + bx"") by 1, without affecting the exponent of x. Method fails when ^ = — 1. TABLE OF INTEGRALS Fundamental Forms x^dx = jL»n+l n+l 'dx , — ^logx e^ dx = (? a^ dx a" \ogea dx l + x'' dx = tan~^ X = sin~^ X Vl- x^ dr. sec~^ X X Vx^ — 1 smxdx = — cosx cos xdx = sin x tan xdx = log (sec x) cot xdx = log (sin x) TABLE OF INTEGRALS 47 / sec xdx = log tan (9 + 2;) / cosec xdx = log f tan - J tan X sec xdx =^ sec x f f f cot :jc cosec X (/x = — cosec x sec^ xdx = tan x cosec^ xdx = — cot x Expressions involving {a + bx) : = 7 log {a + bx) {a + bx) b dx 1 {a + bxY b{a-\- bx) = Tk[(^ + bx — a\og {a -\- bx)] f. J (a + bx) ~ W- dx 1^ a-\-bx log x{a-\- bx) a x dx 1 \ . a+bx x{a-\r- bx)^ a{a + bx) c? x dx \ , b , a-\-bx -— + 72 log x^ {a + bx) ax a^ X Expressions involving (a + bx^) or (a^ zb x^) rfx 1 ,x = -tan~^- a^ + ^^ d d dx 1 , a + ^ = :^log a^ — x^ 2 a a — X 48 ENGINEERING MATHEMATICS r dx 1 , Va+xV-b ., ^f, ,^rt or / , , o = — / log —7^ 7= ifa>0, J<0 J a + bx^ 2V-ab Va-xV^ /dx X 1 r dx (a + bx^^y ~ 2a(a + bx') '^iTaj a + bx^ J a+ bx^ 2b \ bj /dx _ 1 , x^ X {a-\- bx^) 2a a+ bx^ Expressions involving V a + bx: I V a + bxdx = Yl ^(^ + ^^y j x"va jxWa -\-bxdx = 2{2a- 3bx)V{a+bxY 15 b^ -\-bxdx = 2 (8 a2 - 12 a&x+ 15 bH'') V(a + bxf 105 63 dx = 2 \/a -\-bx-\- a j — . ^ J X \ a + bx /dx _ 2 V a + bx Va + bx b /' xdx 2 (2 a— bx) / — r^— / = —TT2 -^cL + bx Va + bx 3 b^ r x^dx 2(Sa^-4abx+3b^x^) /—t-t 1-7= = , g.3 ^-Va + bx J Va+bx 156^ / dx _ J_ Wa +bx— Vg ] X Va+bx Va \Va + bx + Va\ or /dx 2 ^ .. a+bx — , = . — tan-i V xVa+bx v-a ^ ""^ TABLE OF INTEGRALS 49 / dx __ Va+bx _ Jb_ C dx x'^ V a -\- bx ^^ 2 a ,/ ^ V ^ _|_ i)x Expressions involving V a^ — x^ or V a^ + x^: I Va^ — x^ dx = -\x Va^ — x^+ a^ sin~^ - /dx , , X Va^ - x^ ^ r dx _ _ 1 [1 ^+Va[±^l J X Va2 zt ^2 ~ <z L ^ a; J /^^.. = V^^^- a log [^+^^] ±x? fv{a^-x^f dx=^{5a''-2 x^) V^F^^^ + ^a* sin-i - t7 o o Q' J x^\/a^ — x^dx = — ^ V (a^ — x^Y d^ I / OCl + -^ he Va^ — x^ + a^ sin~^ - /x dx X ^ /~o o I ^ • 1 *^ ; = — 7^ V a^ — :r2 + — sin-i - V a^ - ^^2 2 2 a /^^ Va^ - x^ x^ Va^ — x^ a^ a'-x ^ ^ V a^ — rx:^ . - at; dx = sui"^ - X'' X a /Va^ — X x^ Expressions involving V x^ + a^ or V x^ — a^: jVx^±a^dx = i [xVx2±a2±:a21og (x + Vx^ i a^)] 50 ENGINEERING MATHEMATICS I , ^^ = log {x + Va;2±a2] dx 1 , a cos~^ Vx2 - a^ a X 2 -1 ^ J X rV(x2 ± a2)3 </x = I (2 «2 _t 5 ^2) V^^^i^ + ^log(A; + Vr»;2±o2) O Jx dzo; jxWx^zLa'dx = I (2 x2 ziz a^) V:^2 j_ ^2 8 */ V :;c2 dz a^ ^ ^ / dx _ Vx^ =b g^ -^ • = + I0g(x + Vx2=ba2) Expressions involving Vzbax^ -\- bx + c: f , ^^ = ~ log (2 ax+^^+ 2 Va Vax^+te+^j *^ Vax^+bx+c Va TABLE OF INTEGRALS 5 1 /\/ ax^ -\rhx-\- cdx — Vax^ -\-hx-\- c 4a W'-^ac r dx Sa J V'ax^ + bx+c /dx _ I , _^ / 2 ax — b \ V-ax^+bx+c ~ Va ^^^ Wb^ + 4 aJ /V—ax^ + bx+ cdx = V— ax^+ bx+ c 4:a P+4:ac r "a J Formulae involving v 2 ax — x^\ 's/lax — x^ dx ■= — - — V 2 ax— o(?- -\- 1^ sin~^ 2 la // 3 (m ~\~ ax — 2 X / X V 2 ax— x^ dx = 7 V 2 ax — x? b^+4ac r dx 8^ J V- ax^ + bx + c , a^ .X 2 a dx ,x = vers~^ JVlax-x^ '^^^ ^ / . — = — V2 ax — x^ + a vers"^ - V 2 ax - x2 ^ /rfx V 2 ax — x^ X ■\/2 ax - x2 ax ' dx = V 2 ax — x^ + a vers~^ - X <z /Jx _ X — a V(2 ax - x2)3 ~ a2 \/2 ax - x^ /V^^^x = V(a+x)(6+x) + (a- &) log [Va + x + vT+x] 52 ENGINEERING MATHEMATICS I\h^ ^"^ ^ ^^^ - x) (6 + a;) + {a+b) sin-^ \/^ Expressions involving trigonometric forms: /sin^ xdx = ~ — -T^milx) 2 4'' J. ^ , sm^-i ^cos:^,/z— 1 C . „ ^ , sin"^ ^^t: ax = 1 I sin"*""^ x ax w n J j cos^ X rfx = ^ + - sin (2 x) /cos^ X t/x = - cos"""^ X sin X H / cos**"^ n n J X ax I sin X cos X Jx = I sin^ x / sin^xcos^x Jx = —J [|sin (4x) -- x] /cos'^^^ X sm X cos"^ X ax = r-r- w+ 1 sin^ X cos X (/x sin^+i X m+ 1 cos*^ X sin" X Jx = cos"^"^ X sin'^+^ X m + n fft — 1 1 H , — I cos'^-^ X sin** X (/x m + nj cos^xsin'^xt/x = — / / sin^ cos"* sin"~^ X cos'^^^ X + m + n n— \ m-\- n / cos""' X sin'*~2 X dx sin^+^ X - Jx = > ,. , X (fj — 1) cos''~^x , n — m — 2 r sin"^ x ^ H ^ — I — ;r"9- ^^ w — 1 J COS'*"'^ X TABLE OF INTEGRALS S3 / 'cos^ sin'" X , _ cos''+^ X X {m— V) sin'^"^:^ m — n— 2 r cos^ x , m— 1 J sin"'~2 X m — 2 r dx / dx Q,os>x m— 2 r sm^x {m— 1) sin'^~^:\; m— \J \ sin'"~2 X /dx _ sin X I ^ ~ ^ r ^^ cos"" X {n — \) cos"*"^ X n — \ J cos"*"^ x I tan xdx = — log cos x I tan^ xdx = tan x — x j cot xdx = log sin x I cot^ :J[; J:;t: = — cot x — x /sec X ^:. = log tan (I + I) = i log [±|^ I sec^ X J:;t: = tan x j cosec :;t: J:v = log tan (J :x:) /' j x^smxdx = 2xsinx— (x^ — 2) cos a; i X cos xdx = cos a: + ^ sin X I x^ cos :x: (/x = 2 X cos x + (^^ — 2) sin x cosec^ a; fiJx = -- cot x sinxdx = siax — X cos r;t: m+i dx 1 54 ENGINEERING MATHEMATICS Transcendentals \ \ogx dx = xlogx — X rM^"j.=4.(iog.) J X n-\- \ /dx , , J x([ogxY {n — 1) {[ogxY~^ xe'^'^dx = —^{ax— 1) a a J /'^«^ 1 e«^ g r 6«^ j^ ^ ~ ~ m- 1 ^F-i m- 1 J ^^-1 /. . V , r^ sin (;^x) — n cos (wx)l 6«^ sm (/zx) J:r = ^«^ ^ — Yjr~2 ^ — /. . - Va cos (fix) + ;^ sin (/^:;t:)~| 6^^ cos (#x) J:^ = 6«^ ^ ^ ' .3 ^-^^ HYPERBOLIC FUNCTIONS Hyperbolic Transformations ^x ^ — X sinh X = — z = — i sin (/a:) where j = V— 1 cosh :;!£; = = cos [jx) dx ^ HYPERBOLIC FUNCTIONS 55 tanh X = -—-. = — / tan ijx) e + e~^ coth X = — = / cot ijx) e — e e^ = cosh X + sinh x e~^ = cosh X — sinh X sin X = — j sinh {jx) cos X = cosh ijx) Hyperbolic Formulae cosh^ X — sinh^ x = I sech^ X + tanh^ x = I coth^ X — cosech^ x ^ I sinh ix+ y) = sinh x cosh y + cosh ::t: sinh y cosh (x + }') = cosh X cosh y + sinh x sinh y sinh (x — j) = sinh x cosh y — cosh x sinh 3; cosh {x — y) = cosh x cosh y — sinh x sinh y , . . _ tanh ::!:: + tanh y 1 + tanh X tanh y , , , . coth X coth 'V + 1 coth (:x: + ^'j = — -r j — ^-r — ^ coth 3; + coth X , , , V tanh X — tanh v tanh {x — y) = 1 — tanh :^ tanh y - , . coth :^ coth a; — 1 coth {x — y) = — TT ^TT — coth y — coth a; sinh (2 a:) = 2 sinh x cosh x cosh (2 x) = cosh^ x + sinh^ x 2 tanh x tanh (2 x) 1 + tanh^ X 56 ENGINEERING MATHEMATICS C0th2:r+ 1 coth (2 x) = 2 coth X (I) = v/' . 1 f -^ 1 . / cosh X — 1 sum cosh'"> ..cosh^+1 (i)=N/: tanhi;i-V/':5!^ cosh x+ 1 (0 = V ; ..... . , cosh X + 1 coth coshx sinh X + sinh y = 2 sinh ( — y-^ j cosh f — ^ j sinhrJt: — sinhy = 2 cosh I — ^jsinhf "^ j cosh :\: + cosh 3; = 2 cosh f — T-^jcosh( — j^j cosher — coshy = 2 sinh f — j^j sinh f — j-^j sinh (3 :x:) = 3 sinh x + 4 sinh^ x cosh (3 :x:) = — 3 cosh x + 4 cosh^ :;»; Inverse Hyperbolic Functions sinh"^ X = log (x + Vl + x?) cosh"^ X = log {x + Va;2 — 1) tanh-ix=ilog[[±|] HYPERBOLIC FUNCTIONS 57 sech-i X = .o.(i + v/^ ■) cosech"^ : «-log(^+v/?- ^.) Differentials of Hyperbolic Functions d (sinh x) = cosh X dx d (cosh x) = sinh X dx d (tanh x) = sech^ X dx d{cot\ix) = — cosech^ X dx d (sech x) = — sech X tanh x dx d^ (cosech x) = — cosech X coth :x:^x d (sinh~^ x) Jx Vl + x^ d (cosh~^ x) dx Vx^- 1 d(t2Lnh-^x) 1-x^ d (coth-i x) dx d (sech~^ x) rfo; :^Vl-rt;2 d (cosech""^ x V _ dx ) / ^ . . Use of Hyperbolic Functions Illustrative Example. Deduce an expression for the length of a perfectly flexible chain suspended be- tween two supports; assume that both points of sup- port are the same height from the ground. 58 ENGINEERING MATHEMATICS The chain assumes the form of a cate- nary (see page 27), the equation of which IS y = a cosh The general equation for the length of the chain is L = length iV'+(g)'- ciy where the value of -— , obtained by differentiating the (JL% equation of the catenary, is dy dx I a cosh - j dx 4('^"^a) ©. sinh- a dy . Substituting the value of -~ in the formula for the length, Lj we have L =f\i + sinh2^ dx = Ty cosh2 ^ j^ = I cosh - J a dx = a sinh which is the required expression for the length of the chain. DIFFERENTIAL EQUATIONS A differential equation is a relation involving deriv- atives or differentials. A solution of a differential equation is a relation DIFFERENTIAL EQUATIONS 59 between the variables which satisfies the given equa- tion. ORDINARY DIFFERENTIAL EQUATIONS Equations of the First Order and First Degree I. An equation of the form fi(x)dx+f2{y)dy = can be integrated immediately. Its solution is J/i (x) dx + J/a {y) dy = C An equation may sometimes be changed to the above form by separation of the variables. II. Homogeneous Equation. An equation is homogeneous in respect to its variables when the sum of their exponents is the same for each term of the equation. Homogeneous equations are reduced to the form of Method I, by substituting vx for y, and then separating the variables. III. Non-homogeneous Equation of First Degree in X and y. This type occurs in the form: {ax + by + c) dx = {a!x + Vy + c') dy Substitute for x^ {x! + A), and for y, (y' + k). The equation then becomes: {ax'-^ by+ah+hk-\-c)dx' = {a'x'+Vy+a'h+b'k+c')dy Equating ah+ bk+ c = and a'h +b'k + c' =^0 the original equation now takes the form: (ax' + by) dx' = (aV + by) dy which is homogeneous and solvable by Method II. 6o ENGINEERING MATHEMATICS In the solution thus obtained, substitute x' = X — h and y' = y — k where h and k are determined from the two equations: ah+bk + c = Q a'h + 6'yfe + c' = IV. Linear Equation. A Hnear differential equa- tion (of first order and first degree) is of the general form: where P and Q are functions of x alone or constants. The solution of this equation is: yef'''^=J/'''^Qdx + C V. Equations Reducible to the Linear Equation. This type occurs in the form: where P and Q are functions of x alone. The given equation may be written: ^+(l-n)Pv = (l-n)Q where v = 3;"'^+^ This equation is linear in v, and solvable by Method IV. In the solution, resubstitute for V its value 3;-^+!. VI. Exact Differential Equation. An equation of the form Mdx+ Ndy = DIFFERENTIAL EQUATIONS 6 1 is exact if the derivative of M with regard to y is equal to the derivative of N with regard to x. The solution then is: jMdx+ JIn--^ fudo^dy^C where j M dx is the integral of M with respect to x (regarding y as constant), and the term is found by subtracting from N the derivative in respect to 3; of J M dx. The term L^ "~ 7" j M dx\ is integrated with regard to y (considering x constant). The complete solution is then given by the formula above. VII. Integrating Factors. If a differential equa- tion of the form Mdx + Ndy = is multipUed through by a certain expression called an integrating factor, the equation will become exact. It is then solvable by Method VI. (a) When an equation is homogeneous, ^ is an integrating factor. (b) When the condition exists that dM _dN d^ dx = F {x) [an expression containing only x] then e^^^^^^^is an integrating factor. 62 ENGINEERING MATHEMATICS (c) Similarly when dN^_dM ^ ^ = F(y) then e^^^^^^Hs an integrating factor. Equations of the First Order but Higher than the First Degree In the following formulae, ■— will be denoted by p. An equation of first order and of nth degree is of the general form pn _p jlpn-l J^ ^^n-2 ^ . . . ^ Jp ^ f- = where the coefficients A, B, - • - J, K are functions of X and y. I. Clairaut's Equation. When an equation is of the form y = px+f(p) the solution is obtained by substituting for p a con- stant c, y = cx+f{c) II. Solution by Factoring. The given equation may sometimes be resolved into rational factors of the first degree. Each factor is equated separately to ^ero, and its solution found by one of the preceding methods, using the same constant of integration in each case. The complete solution is then the product of the sep- arate solutions. III. Equations Containing only x and p. When an equation is of this type, solve for p, and substitute DIFFERENTIAL EQUATIONS 63 its value j- . The resulting equation can be integrated immediately. IV. Equations Containing only y and p. Solve for p, and substitute its value -~, This equation is immediately integrable. V. Equations Involving x, y, and p. A solution can be obtained by one of the following methods: (a) Solve for x in terms of y and p. Then differen- doc 1 tiate in respect to y, remembering that — = - . ay p The solution of this equation, together with the given equation, constitutes the complete solution. (b) Solve for y in terms of x and p. Differentiate dy with respect to x, and in place of -p substitute its value p. The complete solution consists of the solution of this equation, together with the original equation. dy (c) Solve for ^, and replace it with its value -f^- From this equation it may be possible to obtain a solution. Linear Differential Equations with Constant Coefficients A linear differential equation is of the first degree in the dependent variable and all of its derivatives. The particular integral is the solution of the equa- tion obtained without the introduction of constants of integration. The complementary function is the solution ob- tained by temporarily equating to zero all those terms 64 ENGINEERING MATHEMATICS of the equation that do not contain the dependent vari- able or derivatives thereof. The complete solution is the sum of the particular integral and the complementary function. A linear equation with constant coefficients is of the form: d^v d^~^v d^~^v where the coefficients P, Q, - - • R are constants; and X is a function of x. Replacing — by the symbol D, the equation becomes {D^ + PD^-''+QD^-^+ . . . +R)y = X. Case I. Method of Solution when X = o. Write the given integral in its symbolic form, replacing -^ by D, Then solve this equation for D as if it were an ordinary algebraic quantity. When the roots of the equation (i.e., the values of D) are real, the solution is y = cxe^^'' + c^e^"^ + • • • where ci, ^2, etc., are the constants of integration, and mi, W2, etc., are the roots of the equation. When two or more real roots of the equation are equal, the solution is ^^ = (^1 + c^oo + CzX^ + . . . ) e*^* + • • • where m is the value of the repeated root, and Ci, ^2, ^3, etc., are the constants of integration (introduced in the manner shown in the above equation) and equal in number to the number of times the root m is repeated. DIFFERENTIAL EQUATIONS 65 When the equation has imaginary roots (which always occur in pairs) the solution is y = e^x^ \A cos {a\x) + B sin {a\x)\ + e^"^^ \C cos (^2^) + Z) sin (02^)] + . . . where A and 5, C and Z>, etc., are the constants of integration, and \m\ d= a\ V — l), yjfn^ i 02 V— l), etc., are the complex imaginary roots of the equation. When two or more pairs of complex imaginary roots are equal, the solution is y = [(^1 + ^2:^ + • • • ) cos {ax) + (^3 + ^4^+ • * • ) sin (ax)] 6"^* where (w it a V— l) is the repeated pair of complex imaginary roots. Case II. Method of solution when X is not equal to zero. In this case, the complete solution is the sum of the complementary function and the partic- ular integral. The complementary function is found by tempo- rarily equating X = 0, and obtaining the solution by the method of Case I. The particular integral is obtained as follows. The given equation is of the general form: (£)- + PZ)^-i + (22^^-2+ • • • ■\-K)y = X in which D is used in place of -7- • In symboHc notation, this equation may be expressed The particular integral can then be written: y = Tjjy: = particular integral 66 ENGINEERING MATHEMATICS A. Method of obtaining the particular integral when the term X is of the form e^^. V pCix pCiX particular integral =^_ =^-^ =^^ which is found by substituting the constant a in place of D. This method for evaluating , . fails when the term {D — a) is a factor of / {D). The particular integral is then found by substituting the constant a for D in all terms of / {D) except in the factor {D — a). The solution is then completed by the general method given under case F (page 68). B. Solution for the particular integral when X has the form x^. particular integral = j^ = 77^ = [f{D)]-^x'^ To evaluate this expression, expand [/ {D)]~^ into a series of ascending powers of D, by use of the binomial theorem. It is only necessary to carry out this expan- sion to the mth power of D, since operation on x'^ by higher powers of D would produce zero (since the symbol D stands for — , the operation by Z> on a quantity denotes its derivative with respect to x, the operation by D^ denotes its second derivative, etc.). In obtaining the solution of the given particular integral, x'^ is operated on separately by each term of the expansion of [/ {D)]-^, C. Method of obtaining the particular integral when X has the form sin (ax). X sin {ax) particular integral = / {D) J (D) DIFFERENTIAL EQUATIONS 67 In order to evaluate this integral, substitute — a^ for £)2 wherever U^ occurs in f (D), The particular inte- gral will then be a fraction, whose numerator is sin {ax), and whose denominator is the value assumed by / {D) when D^ is replaced by — a^. This method fails if / (D) becomes zero when — a^ is substituted for D^, The particular integral is then evaluated by writing the term 6'"^ (in which i = V— l) in place of sin (ax). The solution of this new integral is obtained by method A for the evaluation of the particular integral. In the result, ^^''^ is replaced by [cos (ax) + i sin (ax)], producing a result containing both real and imaginary terms. The required particu- lar integral is the coefl&cient of i (i.e., V — l) in this expression. D. Particular Integral when X = cos (ax). The particular integral is obtained as in method C, with the exception that cos (ax) is used in place of sin (ax). When this method fails, ^'''^ is written in place of cos (ax), and this new integral is evaluated by method A. In the solution of this integral, e*""^ is replaced by [cos (ax) + i sin (ax)]. The required particular inte- gral is the real part of this result. E. Particular integral when X is of the form e^^Q. particular mtegral =^^ =^-^ = '^^fWTaj To evaluate the given integral, (D + a) is substituted for D, wherever D occurs in / (D) ; and the term e"* is treated as a constant multiplier. The new integral is evaluated by one of the preceding methods, J(D+a) or by the general method F. The required particular 68 ENGINEERING MATHEMATICS integral is then equal to the product of ^"^ by the evalua- tion of TTTs^-T- F. General method for finding the particular integral. To evaluate jjjyr X The denominator of , . may be resolved into factors of the first degree. The given integral then becomes: 1111 1 {D -a) {D- h) (D -c) {D- d) {D - m) X The term X is operated on successively by each of these fractional operators, beginning at the right. The operation on X by the first factor . _ — r produces the expression e""^ / ^""^^ X dx. This result is operated on in a similar manner by each remaining factor (pro- ceeding from right to left). The solution of the given particular integral is then: Homogeneous Linear Equation The homogeneous linear equation is of the form in which the coefficients P, ... i? are constants, and X is a function of x. DIFFERENTIAL EQUATIONS 69 On assuming the relation, x = e^, this equation may be transformed by the substitutions: x""^ = d{e-l){e- 2) ' ' ' ton terms I ^^^"'£3 = (^- 1) (^- 2) (^- 3) . . . to (^- 1) terms, and so forth; where the symbol d stands for — • The complementary function is then found as in the case of the linear equation with constant coefficients. (In obtaining this solution, the term 6 is treated in exactly the same manner in which the term D was treated in the preceding cases.) i In order to obtain the particular integral, the term X (which involves only x) is changed to an expression involving z, by the substitution x = e^. The particu- lar integral is then found by one of the methods given under the case of the linear equation with constant coefficients. The complete solution is the sum of the comple- mentary function and the particular integral. In the result, z is replaced by its value log x. Exact Diflferential Equations An exact differential equation is one which can be derived directly by differentiation of an equation of the next lower order. jk If the given equation is of the form: where A, B, , . . Q, Rj S, T, and X are functions of X, we then have as the condition for exactness that: 70 ENGINEERING MATHEMATICS ' j,_dS_^<PR_<PQ^ ... =0 dx dx^ do(? The first integral of the given equation then is: dx^-^ "^ V dx Jdx^-^ "^ V ^^ dxydx^-^ = Cxdx + C This formula may be reapplied successively as long as each resulting equation satisfies the condition for exactness. Equations of the Second Order and the First Degree General form is where P, Q, and X are functions of x, I. When one solution of the equation is Imown (or can be found by inspection). Let yi equal the known integral. In the given equation, substitute vyi in place of y; and then, in the dv transformed equation, replace — by p. This equation can be solved by one of the preceding methods. II. Change of the Independent Variable. The purpose of this change and of the removal of the first derivative (see III) is to transform a given equation into a new equation which may happen to be easily integrable. The given equation is of the form: DIFFERENTIAL EQUATIONS 7 1 By changing the independent variable, it may be transformed into the following equation: where Qi becomes equal to 1, if 1=^5 when also dh dz „ dx''^ dx Y ^ and Q or where Pi may be made equal to zero, if e"*^ ^dx -f when also Q\ = -jTr"^ dx) X and Xi = \dx) III. Removal of the First Derivative. To remove the first derivative from an equation of the form f| + P^+(3y = Z dx^ dx ^^ make the substitution y = ve~^'^ ^ The given equation then becomes where Q,=g-lf_lp3 and Zi = Ze*-/'^'^ 72 ENGINEERING MATHEMATICS THEORETICAL MECHANICS Center of Gravity The center of gravity of a body is a point so situ- ated that the force of gravity produces no tendency in the body to rotate about any axis passing through this point. Center of Gravity of the Arc of a Plane Curve Jxds Jx\/l + (£jdx . £* /Vi+(|)V where x and y are the coordinates of the center of gravity. Solve for y in terms of x from the equation of the dv given curve. Then differentiate in order to obtain -p, and substitute its value in the formula for x. dx Similarly, find x in terms of y, obtain — , and sub- stitute in the formula for y. Center of Gravity of Plane Areas Rectangular Coordinates : I J xdA I j xdxdy X = JJdA ffd.dy THEORETICAL MECHANICS 73 I j ydA I j ydxdy y = I I dA j j dxdy where x and y are the coordinates of the center of gravity. In evaluating the expression for x, we may integrate first either in respect to x or y, according to which method is more convenient. If dy is integrated first, the Umits of y are expressed in terms of x (from the given equation) ; and the Umits of X are its initial and final values. Similarly, if dx is first integrated, the limits of x are expressed in terms of y; and the limits of y are then its initial and final values. Polar Coordinates : / I r^ cos 6 dddr X = ^ ^ u If rdddr r^siaddedr If' rdddr Generally, it is more convenient to integrate first with respect to r. In this case, the Umits of r are found in terms of 6 from the equation of the given curve. The Umits of 6 are its initial and final values, expressed in radians. Center of Gravity of Solids of Revolution. When a solid of uniform density is formed by the revolution of a plane curve about the X-axis, the center of gravity 74 ENGINEERING MATHEMATICS is on the X-axis (because of symmetry). Its x-coordi- nate is X = I I xydx dy j I y dxdy where the limits are found as in the case of plane areas. When a solid is formed by the revolution of a plane figure about the F-axis, the y-coordinate of its center of gravity is / j xydx dy I I X dxdy Center of Gravity of Any Section Composed of Two or More Simple Plane Figures In order to find the center of gravity of such figures as tee-bars, channels, rails, etc., divide them up into their component rectangles or triangles. Then, obtain the center of gravity and the area of each separate figure. Choose any convenient axis in the plane of the given section and find the turning moment of each figure about this axis. Each turning moment is the product of the area of the figure by the distance from its center of gravity to the chosen axis. The sum of all these separate turning moments gives the turning moment of the total figure. On dividing this total moment by the total area of the figure, we obtain the distance from the chosen axis to the center of gravity of the figure. Care must be used, if the chosen axis passes through the given figure, to take distances on one THEORETICAL MECHANICS 75 side of this axis as positive, and on the other side as negative. Generally, one coordinate of the center of gravity can be determined by the symmetry of the given sec- tion. When the figure is unsymmetrical, it may be necessary to take moments about two different axes in order to locate the center of gravity. Moment of Inertia of Plane Areas The moment of inertia of a plane figure about any given axis is equal to the integral of the product of each elementary area of the figure by the square of its distance from the axis. Rectangular Moment of Inertia: The rectangular moment of inertia of a plane figure is its moment of inertia about any axis in the plane of the figure. The rectangular moment of inertia of a plane area about the X-axis is = //: y^ dx dy The rectangular moment of inertia of a plane area about the F-axis is I = I j x^ dxdy In either case, the limits of the variable first integrated are expressed in terms of the other variable. The moment of inertia of a plane figure about the gravity axis (Ig) is its rectangular moment of inertia about any axis in the plane of the figure, passing through its center of gravity. The moment of inertia of a plane figure about any axis parallel to the gravity axis and in the plane of 76 ENGINEERING MATHEMATICS the figure is equal to {Ig) plus the product of the area of the figure by the square of the distance between the two axes, thus: I =Ig + Fd? Polar Moment of Inertia: The polar moment of inertia (Zp) is the moment of inertia about any axis perpendicular to the plane of the given figure. It is equal to the sum of the rectangular moments of inertia about two mutually perpendicular axes in the plane of the figure, passing through the foot of the polar axis. In rectangular coordinates, the polar moment of inertia equals /p = /. + ly =//(^' + f) dxdy In polar coordinates, the formula for the polar moment of inertia is -// B?dRde It is generally more convenient to integrate first with respect to R, expressing its limits in terms of d. The limits of 6 are then its initial and final values. Moment of Inertia of Solids The moment of inertia of a solid (with center at origin) about the X-axis is I = m j j I (y'^+ z^)dxdydz where m is the density, that is, the mass per unit volume. THEORETICAL MECHANICS 77 Radius of Gyration The center of gyration is that point in a revolving body at which, if the entire mass of the body were concentrated, the moment of inertia about the axis of rotation would be the same as that of the body. The radius of gyration, k, is the distance from the axis of rotation to the center of gyration. For plane sections . *=v/-! For solids, k.\jL. /X in which k = radius of gyration, / = the moment of inertia about the axis of rotation, A = area of section, M = mass of body, W = weight of body. Center of Percussion The center of percussion or oscillation of a pendulum or other body vibrating or rotating about a fixed axis or center is that point at which, if the entire weight of the body were concentrated, the body would continue to vibrate in the same intervals of time. The radius of oscillation is h = Md iiy 78 ENGINEERING MATHEMATICS in which I = the moment of inertia of body about axis of rotation, d = distance from center of gravity of body to the axis of rotation, h = distance from center of percussion or oscillation to the axis of rotation, M = mass of body, W = weight of body. Motion of a Body velocity at any instant = v ds di . , . dv d^s acceleration at any mstant = ^ = 37 = df In rectangular coordinates, dx _ ds dt ~di Vx = ^7 = -j: cos d = velocity in a direction parallel to the X-axis dy ds ds /(dxV (dyV ' = d-r\[dt)+[dt) For motion with uniform velocity, Y */ ^. ^ — 1 J V = J For uniformly accelerated motion, s = ^ {u + v)t s = ut + ^ at^ 2 as = v^ — u^ THEORETICAL MECHANICS 79 u = initial velocity, V = final velocity, a = constant acceleration, s = space passed over, / = time of motion. If the body starts from rest, the initial velocity u equals 0, and these equations become: s = ^ vt s = \ af 1 as = 1? Rotation of a Rigid Body velocity at any mstant = ^ = ;77 acceleration at any instant = a = -rr = "tto ^ dt dt^ For motion with uniform velocity, e For uniformly accelerated motion, ^ = i (coo + co) / d = angular space through which the body rotates, coo = initial angular velocity, CO = final angular velocity, a = angular acceleration, t = time. 8o ENGINEERING MATHEMATICS For a body initially at rest, the velocity coo is 0, and these equations become e = ^ o)t e = ^af Falling Bodies Equations of motion of a body falling from rest under the action of gravity: V = gt 2 gs = v^ V = velocity after time /, 5 — height through which body falls, g = (approx.) 32.16 feet/sec.^ = 981 cm/sec.^ = acceleration of gravity. The value of g for any latitude and any altitude is g = 32.0894 (1 + 0.0052375 sin^^) X (1- 0.0000000957 £) in which 6 = latitude of place in degrees, E = elevation above sea-level in feet. Projectiles Equations of a body projected vertically upward with an initial velocity u (resistance of air not con- sidered) : (1) Velocity at any time = u — gt, (2) Velocity at any height = V w^ __ 2 gh. (3) Height at any time = ut^ \ gf. THEORETICAL MECHANICS 8l (4) Greatest height = — . (5) Time of flight = — . o Equations of a body projected with an initial velocity w at an angle 6° to the horizontal (resistance of air not considered) : The curve described by the projectile is the parabola whose equation is y = X tan B — lu'cos^e where B is positive when the body is projected above the horizontal and negative when the body is projected below the horizontal. Horizontal-component of acceleration = "T^i = Vertical-component of acceleration = -^^ = ^ g (1) Velocity at any time = \^u^ — 2 utg sin B +g^^^. (2) Velocity at any height = Vw^ _ 2 gh, (3) Height at any time = utsinB — ^ gt^. /A\ rr.' r n- t-^ 2wsin^ (4) Tune of flight = . rc\ -n u^ sin (IB) (5) Range = ^ — -. o If the friction of the air is taken into account, the curve described by the projectile is given by the empirical relation: gx^ / 1 , kx\ k = 0.0000000458- w - I 82 ENGINEERING MATHEMATICS where d = diameter of projectile in inches, w = weight of projectile in pounds. Angular Measure A radian is the angle subtended at the center of any circle by an arc equal in length to its radius. 180 1 radian = degrees = 57.296+ degrees TT 1 degree = 7^7: radians = 0.0175+ radians loU The relation between the central angle of a circle and its subtended arc is given by the formula: l = rd I = length of arc, r = radius of circle, 6 = central angle in radians. Circular Motion A body moving with uniform velocity in a circular path experiences a constant acceleration toward the center of the circle. This acceleration is expended in changing the direction of motion of the body. The equations of motion of the re- volving body are a = — r vT = 2 7rr 47rV ^ = -f2- V = constant velocity of particle in feet per second, a = constant acceleration toward center in feet per sec.^, THEORETICAL MECHANICS 83 r = radius of circular path in feet, T = time of 1 revolution in seconds, 7r2 = 9.8696+. If the body moves with a variable velocity, then: tangential acceleration = -r. normal acceleration = — Centrifugal Force The centrifugal force of a revolving body, in pounds, is p ^Wf^ ^ 47rWr gr ~ g/2 or in terms of the number of revolutions, iVi, per minute F = 0.00034 WrNi^ W = weight of revolving body in pounds, V = velocity of body in feet per second, t = time of 1 revolution in seconds, r = distance from axis of rotation to the center of gravity of the body, in feet, g = acceleration of gravity (32.16). Flywheel The energy of rotation of a flywheel is K.E. =^ = IttHN^ I = polar moment of inertia about the axis of rotation, CO = angular velocity in radians per second, N = number of revolutions per second. 84 ENGINEERING MATHEMATICS The energy stored in a rim flywheel by a variation in speed is W E = ~ (S^max — S^min) foot-pOUnds, s W = weight of flywheel in pounds, •Smax = maximum rim speed in feet per second, Sndn = minimum rim speed in feet per second g = acceleration of gravity (32.16). The rim speed in feet per second is 5 = IwRN, where N is the speed in revolutions per second, and R is the radius of the wheel in feet, measured from the center of gravity of the rim section.* Hence, the energy stored is E = ^ foot-pounds and the weight of the flywheel is W= ^ Substitute for E the required stored energy in foot- pounds. Assume some convenient value for R, in feet; then solve for the weight W in pounds. If the rim speed is too high (average about 35 feet per second for cast iron or 150 feet per second for steel), the value of R must be reduced. The ratio of the speed variation, iVmax — iVmin, to the average speed may be taken as follows for different types of machines: Hammers 0.20 Punches 0.05 Ordinary machinery 0.03 Textile and paper machinery 0.02 Electric generators 0.005 * This value of R is approximately correct. The exact value of R is the radius of gyration of the flywheel. THEORETICAL MECHANICS 85 Simple Pendulum The time of oscillation in seconds from one extreme position to the other is ^ g I = length of pendulum in feet, g = acceleration of gravity (32.16 approx.). The period of the pendulum is ^ g The seconds-pendulum makes one oscillation per second from one extreme position to the other; its length in feet is Work and Energy For a uniform force, W F = ma = — a g W Ft = mv = ^v g FS =\mi? = -;y- ^ g F = constant applied force in pounds, a = constant acceleration in feet/sec.^, m = mass of body, W = weight of body in pounds, V = velocity acquired after t seconds, mv = momentum, s = space passed over in feet, g = acceleration of gravity (32.16 feet/sec.^). 86 ENGINEERING MATHEMATICS The impulse I of the constant force F during the time t equals the change of momentum, I = Ft = mv — mu where u is the initial velocity and v the final velocity. If the force is variable, then impulse equals =x Fdt The work done by a uniform force is W = Fs = imv^ The work done by a variable force equals W=rFds The kinetic energy of a body of mass m, moving with a velocity v, equals i mv^. Direct Central Impact For the impact of two bodies of the same material, weighing respectively W and Wi pounds, the velocities after impact are Wu + WiUi — eWi {u — ui) V = Vi = W+Wi Wu + WiUi + eW (u — ui) W+Wi ' u = original velocity of W in feet /second, V = velocity of W after impact, Ui = original velocity of Wi, Vi = velocity of Wi after impact, e = coeflQcient of restitution. THEORETICAL MECHANICS 87 Values of e, the coefficient of restitution, for different materials are as follows: glass on glass e = .94 ivory on ivory e = 0.81 cast iron on cast iron e = 0.66 lead on lead e = 0.2 The sum of the momenta of two bodies after impact equals the sum of their momenta before impact, Wv , WiVi g g Wu WiUi g g Two inelastic bodies after impact move with a common velocity _ WiVi + W2V2 ^ " W1 + W2 in which Wi = weight of first body, W2 = weight of second body, Vi = original velocity of first body, V2 = original velocity of second body. Composition and Resolution of Forces The resultant of the forces Fi and F2 acting at a point is R = VFi^ + 2 F1F2 cos (9 + F2' in which d is the angle in degrees between the two forces. The direction of R is determined by the relation F2 sin d tan a = Fi + F2Cosd 88 ENGINEERING MATHEMATICS in which a is the angle in degrees between Fi and R. The rectangular components of a force R acting in a given direction are X = RcosB Y = RsinS in which X is the horizontal component of R, Y is the normal component of Rj and d is the angle in degrees between R and X, The resultant of several forces acting in different directions at a point is R i R = VX^+ F2 in which X = Fi cos Oi + f 2 cos $2 +FzCose,+ . • ' , Y = Fi sin 01 + F2 sin 62 + F3sin^3+ • •. • , where Fi, jF2, F3, etc., are the given forces, and ^1, 62, ds, etc., are the angles in degrees between the given forces and the horizontal axis. Friction F = friction in pounds, N = normal force in pounds, / = coefficient of friction. F = fN Angle of friction = </> = tan" :^ ■i| = tan-V THEORETICAL MECHANICS 89 Average values for /, the coefficient of friction, for motion are as follows: Character of contact / Wood on wood Metal on wood Metal on metal, dry Metal on metal, lubricated Leather on metal, dry Leather on metal, lubricated 0.25-0.50 0.50-0.60 0.15-0.24 0.075 0.56 0.15 Belt Friction P and Q are the forces at the ends of the belt, P being the greater force. F= resultant force of friction, N = normal reaction of pulley, 6 = angle in radians sub- tended by the arc of contact, / = coefficient of friction. or in common logarithms logio^ = 0.434/^ The value of / varies from 0.15 to 0.6 depending on the condition of belt and pulley, but, in general, it is approximately correct to assume/ = 0.3. Inclined Plane Equations of motion of a body sliding down an incline under the action of its own weight. 90 ENGINEERING MATHEMATICS For a frictionless plane: dh (1) acceleration along plane = a = — = g sin0, dt (2) velocity after t seconds = tg sin 6, (3) velocity at bottom of plane = ^/l gh, (4) distance traveled in t seconds = -^-^ — , (5) time of sliding down plane =vi For an inclined plane with friction d^s (1) acceleration along plane = ^ = 3^ = g [sin^ — fcosd], in which / = coefl&cient of friction. Conditions for the equilibrium of a body resting on an incline: W = weight of body, F = applied force, N = normal pressure on plane, 6 = inclination of plane in degrees, / = coefl&cient of friction. MECHANICS OF MATERIALS 91 For a frictionless plane: (1) When the balancing force is applied parallel to the inclined plane, F = Wsmd N = WcosS (2) When the applied force acts horizontally, F = IF tan (9, N=^W seed. For an inclined plane with friction: (1) When the balancing force acts parallel to the incline, ^ TF sin (61=1= (90 cos (^0 in which e' = tan-V (2) When the applied force acts horizontally, F = WtSin(dzL 6') MECHANICS OF MATERIALS Stress is distributed force; its intensity per unit area is generally expressed in pounds per square inch. The elastic limit of a material is the maximum stress in pounds per square inch that will be followed by a complete recovery of form, after the removal of the stress. Permanent set is the change in form of a member when stressed beyond its elastic limit. The ultimate strength of a material is the least stress in pounds per square inch that will produce rupture. 92 ENGINEERING MATHEMATICS Modulus of elasticity is the number obtained by dividing the actual stress in pounds per square inch by the corresponding elongation per inch. The factor of safety is the factor obtained by divid- ing the ultimate strength by the actual stress in pounds per square inch. Tension and Compression For direct stress, uniformly distributed, p = stress in pounds per square inch, P = total load in pounds, F = cross-sectional area in square inches. P F_Pl I E = modulus of elasticity in tension or compression, / = length of member in inches > € = elongation per inch length, X = total elongation in inches. MECHANICS OF MATERIALS 93 CO < P^ W < O o CO O O LO "^ rt '--< lo i^ O CS CM ---H 'D'd OS rt <L> O lOiO 00 o oo»o O OsOOO 'O"* TtH lO oo o o oo oo o vO O «N CN s o o rt tt w'-^ 888 o^o^^o^ CM vo i-T 8 oo o o'd^d^d" SS88 to lO O O ^ CN CO CO O o COOOO o o th rti LO 00 QOO ooo o^o^o^ rovO 00 OOOO OOOO OOOO OMO so CM OOOO OOOO o^o^o^o^^ o^o'^o'^o" CN lO O O CO 8888 O^O^O^O^ vO to LO O CM co»0 sovq CMCMO d vC CMX:^ 00 00 CO &iO CO O O o 03 OJ d fe.s C ^^ .-4 < . <^ t . -^^ •ggss Vh j_> 'f-l -r-l MS ^ CD O w O ? o^ OTCO 94 ENGINEERING MATHEMATICS Angular Distortion and Shear Shearing stress, uniformly distributed equals Ps=p P = load, F = area. For torsion: E. =^ Es = modulus of elasticity in shear, b = angle of distortion in radians. Note. The modulus of elasticity in shear is f as great as in compression or tension. b = Pa = Torsion of Circular Shafts Ps = eotEs p / oilpEi ea T pjp e ^^ 32 Pa='^ ^^ 16 lirPaN Horsepower = jj^^qq^^2 b = helix angle of distortion in radians, a = radial angle of distortion in radians, / = length of shaft in inches, e = radius of shaft in inches. MECHANICS OF MATERIALS 95 p^ = greatest shearing stress in pounds per square inch existing in shaft, Eg = modulus of elasticity in shear, Ip = polar moment of inertia of circular section (see table of standard sections), P = force in pounds producing torsion, that is, the turning force, a = lever arm of force P in inches, d = diameter of shaft in inches, N = revolutions per minute. In deriving the above formulae, the torsion is treated as due to a couple of the same turning moment, Pa, as the single force P with lever arm a. This eliminates the consideration of any stresses other than shearing stresses, and, in applying these formulae to the case of a single driving force, bending stresses and bearing friction are neglected. Flexure of Beams When a beam is strained by a vertical load, the greatest strain will be in the extreme upper and lower fibers of the beam. The intensity of the strain that can be borne by the extreme fibers is the limit of the strength of the beam. The upper fibers are com- pressed and the lower fibers are stretched when a beam is loaded between supports; the converse holds when it is loaded beyond supports. Somewhere along or near the center of the beam the fibers are neither extended nor compressed; the plane of these fibers is called the neutral surface. The line of intersection of the neutral surface with any cross-section of the beam is the neutral axis of the section. 96 ENGINEERING MATHEMATICS If the stresses remain within the elastic Umits of the material in both tension and compression, and pro- vided the modulus of elasticity is the same for both kinds of stress, then the neutral axis of the section passes through its center of gravity. The elastic curve is the curve assumed by a beam under load. The bending moment for any section of a beam is the algebraic sum of the moments of the external or appUed forces acting on the beam on one side of the section. Thus, for the beam shown, the bending moment about A is M = Rix- Pa < The bending moment, M, of any section is nu- merically equal to the Ri R2 moment of resistance of the section, which is the resistance which the particles of the beam offer to distortion. The moment of resistance equals PL e 1 ' i jA ' ! '^ = M = bending moment p = stress per unit area at the outermost element of the section, e = distance of extreme element of beam from neutral axis, / = rectangular moment of inertia of beam section about its horizontal gravity axis. In designing the proper cross-section for a beam, the maximum bending moment (given for standard cases MECHANICS OF MATERIALS 97 pi I under Beam Loadings) is equated to ^— -. The term -, called the section modulus, may be obtained from the table of standard sections of beams. The value of p must not exceed the maximum allowable stress per unit area for the material of the beam. The maximum allowable stress equals the ultimate strength divided by the factor of safety. The equation of the elastic curve and its radius of curvature may be found ^rom the relations: ^, pi EI j^jd^y / X ilf =^=--=£/^(approx.) E = modulus of elasticity of material of beam in tension or compression, p = radius of curvature of the elastic curve, {x, y) = coordinates of any point on the elastic curve. The deflection of a beam at any point is obtained by substituting, in the equation of the elastic curve, the particular value of x in question, and solving for the corresponding value of y, which equals the deflection. The maximum deflection occurs at the section for which ^ = 0. ax Shear The vertical shear in a beam is equal to the first derivative of the bending moment in respect to x, thus Vertical shear = J = -^— ax where M is the bending moment (expressed as a func- tion of x). The value of the vertical shear for any particular 98 ENGINEERING MATHEMATICS section is found by substituting the corresponding value of X in the expression for -7— . The result is the re- quired vertical shear. The maximum bending moment is found by equat- ing -7— = 0, and then solving for the corresponding value of X, This particular value of x is substituted in the equation of the bending moment,' ikf, and the resulting expression equals the maximum bending moment. The horizontal shear in a plane parallel to the neutral surface (that is, the surface in which neither tension nor compression occurs), and at a distance z" from it, equals X (in pounds /sq. inch) = -77^ I zdF where / = total vertical shear in pounds, y" = width of beam section at z" in inches, / = rectangular moment of inertia of entire sec- tion about the horizontal gravity axis, Jz dF = area in square inches of that portion of the section above 2" multiplied by the dis- tance in inches of its center of gravity above the neutral axis. Beam Loadings M = bending moment, Mm = maximum bending moment, y = deflection at any point, d = maximum deflection, P = concentrated load, W= uniformly distributed load. MECHANICS OF MATERIALS 99 Cantilever Beam with Concentrated Load at the Free End M = P{l-x) Mn.=Pl <A ^P_ll^_ ^ d = EI\2 PP 3 EI m^ f shear moment M = Cantilever Beam with Uniform Load W{1- xf 21 Mr Wl Wx^{ll^+{11-Xf\ R l^EIl d = 8 EI lOD ENGINEERING MATHEMATICS Beam Supported at Both Ends and Loaded with a Concentrated Load at Center M = -7yX Px (3/2-4 x^) 48 £/ d=- 48 EI n I R ^ R T shear 1 R moment Beam Supported at Both Ends and Uniformly Loaded M = Wx {I — x) 21 y//////^^//////A yr _Wl y = Wx {P -2lx^ + X?) 24 Ell d = 5WP 384 EI moment^ MECHANICS OF MATERIALS lOI Beam Supported at Both Ends and Loaded at Any Point i-X-«i ' 1 i : ^ 1 ^■^•^ ^bJfK^ R,l shear .R2 M = x<a ,, Pbx „, V M = -7 P{x — a) x>a M, Pab I y^zwi^'^^"-''"-'^ x< a Pa {I— x) .^ J 2 2\ 'v = ^ (2 Ix — x^ — a^) ^ 6 Ell ^ ^ x> a Pb d = ,,^y3{2ab + a^y 27 Ell occurring when x = i v 3 (2 ab + a^) I02 ENGINEERING MATHEMATICS I Beam Supported at Both Ends and Loaded with Two Concentrated Loads at Equal Distances from Each End 1 i*- a J t a -J ^ -± 1 1 , i ' 1 J R --^ , T« • " n E shear R r \ f Mm / moment M =Px M = Pa M^ = Pa _ Px ^ Pa ^ 6£/ (2>lx-Zx^- a^) '^ 6ElW aA x<a x> a x< a x> a MECHANICS OF MATERIALS 103 Beam Fixed at One End, Supported at the Other, and with a Concentrated Load at Any Point ^°4 W Ri X'; A4^ ]R, shear i?i = Pa2 (3 / - a) 2^ R2 = P-Ri M = P(a-x)-Ri(l-x) M = Ri{x-l) M„. = Ri{l- a) 1 y = y = 6 EI 1 6£/ (i?i:!c» - 3 i?i?x2 + 3 Pax^ - Po^) {Riofi - 3 i?ik2 + 3 Pa^^ - Pa?) 6 EI occurring when '-'i^-^i^J x< a x> a x< a x> a I04 ENGINEERING MATHEMATICS Beam Fixed at One End, Supported at the Other and Uniformly Loaded y/}iiniiiiiii}^^niimiiK "moment J?l = iw i?2 = iw M = 4:X){1-X) ^« = Wl 8 y = Wx" AS Ell {l-x){Sl- 2x) d = ■■ 0.0054 EI occurring when x = 0.5785/ MECHANICS OF MATERIALS 105 Beam Fixed at Both Ends and Loaded at the Center i : 1 ^ shear 1 Mmj\^ ft Mm flioment M = |(4x-0 M™ = — 8 <Z = 48 £/ 192 EI io6 ENGINEERING MATHEMATICS Beam Fixed at Both Ends and Uniformly Loaded w y = wi 12 24 Ell WP 384 EI Q - xy COLUMNS Note. The breaking load in Euler's and in Gordon's formula, and the safe load in Ritter's formula are in pounds. In all of the formulae for columns, the length, I, and radius of gyration, k, must be expressed in the same units (generally inches). Euler's Formula (1) Column with round ends, breaking load = El'^=ir^EF (pj tm MECHANICS OF MATERIALS 107 (2) Column with flat ends, breaking load = 4£/^ = ^ir^EF (^j (3) Pin-and-square column (column with one end round and the other flat), breaking load = -EI j^ = -w^EF Ij^j in which E = modulus of elasticity of material of column in tension or compression, / = rectangular moment of inertia of cross-section about neutral axis, I = length of column, F = area of cross-section in sq. inches, k = least radius of gyration of section. Gordon's or Rankine's Formula (1) Column with flat ends, breaking load = - 2 1+ "■" HI) (2) Colunrn with rounded ends, FC breaking load = rrrg (3) Pin-and-square column, FC breaking load = rrr^ io8 ENGINEERING MATHEMATICS in which F = area of cross-section in square inches, C = ultimate compressive strength of material of column in pounds per square inch, I = length of column, k = least radius of gyration of section, ^ = empirical constant. Values of /5 and of C, in Gordon's formula, are as follows for different materials: Material Hard steel Medium steel Soft steel Wrought iron Cast iron Timber C (lbs./ sq. in.). 70,000 50,000 45,000 36,000 70,000 7200 ^ 1 1 1 1 1 6400 1 25,000 36,000 36,000 36,000 3000 Ritter's Formula (1) Column with flat ends, FC safe load = 1 + c /iv (2) Column with rounded ends, FC safe load = i + ^C-Y (3) Pin-and-square column, safe load = FC 1 + 1.78 C /IV 47r2£ \k MECHANICS OF MATERIALS IO9 in which F = area of cross-section in square inches, C = maximum safe compressive stress of material of column in pounds per square inch, C = compressive stress at elastic limit in pounds per square inch, E = modulus of elasticity for tension or compression, I = length of column, k = least radius of gyration. J. B. Johnson's Formula Breaking load in pounds; cross-section in square inches. For mild steel: (1) Pin-ends, breaking load = [42,000 - 0.97 {t)^F 7 ) not > ISO (2) Flat ends, ^ = I 4-9 nnn — n <S9 / breaking load = [42,000 - 0.62 f^V \F (e not > 190 For wrought iron: (1) Pin-ends, breaking load = [34,000 - 0.67 (i)'\f I , ,not>170 no ENGINEERING MATHEMATICS (2) Flat ends, breaking load = [34,000 - 0.43 (r)^^ (|)not>210 Notation same as in Ritter's formula. Straight-line Formula Breaking load in pounds; cross-section in square inches. For mild steel : (1) Hinged ends, breaking load = [52,000 - 220 (^\]f (2) Flat ends, breaking load = [52,000 - 179 (^'If For wrought iron : (1) Hinged ends, breaking load = [42,000 - 157 (~\ F (2) Flat ends, breaking load = [42,000 - 128 (^ F Notation same as in Ritter's formula. Wooden Columns The breaking load in pounds for solid wooden col- umns with square ends is (700+ 15 m)FC P = 700+l5m + m^ MECHANICS OF MATERIALS III F — cross-section in square inches, m — ratio of the length, /, of the column to the least di- mension (/, of the cross-section ( that is, w = ^ j, C — ultimate compressive strength of material of column in pounds per square inch. Values of C, the ultimate compressive strength, for different kinds of timber are as follows: White oak and Georgia yellow pine 5000 Ib./sq. in. Douglas fir and short-leaf yellow pine . . . 4500 Ib./sq. in. Red pine, spruce, hemlock, cypress, chest- nut, CaUfornia redwood, and Cali- fornia spruce 4000 Ib./sq. in. White pine and cedar 3500 Ib./sq. in. The proper factor of safety for yellow pine varies from 3.5 to 5, according to the amount of moisture present in the timber, being greater for larger amounts of moisture. For all other timbers, the proper factor of safety varies from 4 to 5. CENTERS OF GRAVITY Plane Figures Triangle The C.G. is on a median line of T the triangle, two-thirds of its length j^ from the vertex. ^ = 3 Parallelogram The C.G. is at the inter- section of the diagonals, - h y-2 112 ENGINEERING MATHEMATICS y = Trapezoid 3 (a + 6) Quadrant of Circle _ 4r _ ^ = 31^ = ^ r = 4 rV2 Quadrant of Ellipse _ 4a ^-. 4& ^ = 37 Semicircle - 4r ^ = 3^ 4 r Circular Sector _ 2 re y = 3a MECHANICS OF MATERIALS 113 Circular Segment - — 4 r sin^ d ^~ 3 2 (9 - sin (2 Q) is in radians Sector of a Circular Ring X — IZ 2R^ -r^sind SR'-r^ d Parabolic Segment _ 3h Parabolic Segment x = ib ir I h. 114 ENGINEERING MATHEMATICS Solids Hemisphere Right Pj^amid or Cone - h y-i MECHANICS OF MATERIALS IIS MOMENT OF INERTIA OF SOLIDS W M=mass of body= — Shape of figure ^^ r2r^ ^2R Descrip- tion uniform thin rod thin rec- tangular plate thin circular plate solid cylinder, radius, r hollow- cylinder, i?= outer radius, r= inner radius solid sphere, r= radius hollow sphere, jR= exter- nal ra- dius r= inter- nal ra- dius Axis of rotation (1) through center perpendicular to length (2) through end per- pendicular to length (1) through center of gravity per- pendicular to plate (2) through center of gravity paral- lel to side b (1) through center perpendicular to plane (2) any diameter (1) axis of cylinder (2) through center of gravity, perpen- dicular to axis of cylinder (1) axis of cylinder (2) through center of gravity per- pendicular to axis through center through center Moment of inertia /2 M- M M M 12 12 M (^i) M 'R2+y "(n* .) ^2+^ M 2r^ M 2 / J?5-y5 X 5 \R^-r^) ii6 ENGINEERING MATHEMATICS >> 1 s fe r> o +j 0) c -t-> <u a B ?^ o S ^ o ^ K.^ |vO ^H rO + W •^ -'^ ^ •-^ + ^C> n^ fri 1 f^ >^^ •^ ►«|(N /"^ »<:>l^ fC^ C^ '%^ II (DO «o S.1VO Sb 11^ V) rt Q C^ <4-l ^ O CO ^N S.I2 ^12 CN ^12 ^IS SP CN CM \t << M <5 ►55 I ^0 ^0 11^ ■^^ .C^ K^ ^ m MECHANICS OF MATERIALS 117 I CO + Nl'* I -Td •^ '^ I CO vO ^12 % OS s N -IS I ^0 1^ O 10 o ^ SH s^ VO 00 rCi vO (N 1 00 00 1 . •^ ►o Vii t<-j=-^ 'Ff ii8 ENGINEERING MATHEMATICS o \—{ O P < m O »— I P^ Ph o ss -IS + eo -12 r^ eo I ^^1 0) T3 ^' 6 ^ •«-> d II 1^ _ d 'd rmine page? tie mo ^ .S dete n on s. T ^11 +^ W> £>. ^ O 5i canno ethod gravit o ^ 1 l-o y of which is by the m s along the 1 o 11 o7> ravit ty aJi base 1 Xf\ en tj d t of g] gravi their -o ^•§ t 6 ::; center izontal s, with |1 ^1^ ■o 1 II ?. o ngle-bars, ion of the nent recta 3 .52 § cC .ti O <i> ^11 •4J rt ., o >. earns nd the its c al ^ i2 g asT-b irst, fi ion int 3 2 «« «r sections oUows: F e the sect CO d 2i i2 .2 S) ia of such ained as f hen divid -3) inert obt . T d £1 O M <n xn rj oment of , may b e Section S o-^ _c3 3 'a (U -f-J o Ui o ^? sx ^. ^ (U ■^ QJ H ^ a Pi <u-ti ^ .So O H ;=! HYDRAULICS II9 HYDRAULICS Head and Pressure The difference in level of water between two points is called the head. The pressure in pounds per square inch at any depth is p = 0.433 h in which h = head or depth in feet of water, 0.433 = weight of a column of water 1 foot high and 1 inch in cross-section. The pressure on a submerged surface is always normal to the surface, and equals P (in pounds) = 0.433 hF h = depth of water in feet from the surface of the liquid to the center of gravity of the sub- merged surface, F = area of submerged surface in square inches. Center of Pressure The center of pressure of a submerged surface is the point of application of the resultant of all the fluid pressures on such surface. The distance of the center of pressure of a vertical submerged plate below the liquid surface is d (in feet) = ~ 120 ENGINEERING MATHEMATICS F = area of plate in square feet, i = distance in feet from the liquid surface to the center of gravity of the plate, la = rectangular moment of inertia of plate about the line of intersection of its plane with the surface of the liquid. The distance of the center of pressure of a sub- merged plate inclined at an angle d with the surface is d (m feet) = -^-^ h z rz z = distance from the liquid surface to the center of gravity of the plate in feet, F = area of plate in square feet, Ig = moment of inertia of plate about its gravity axis parallel to the liquid surface. Flow through Apertures Due to friction, the velocity of discharge through an aperture in a thin plate or plank is reduced about 3 per cent below its theoretical value. Further, on leaving the orifice, the jet contracts to approximately 64 per cent of the area of the aperture. The theoretical velocity of discharge through a small aperture, in feet per second, is ^ = VYgh g = acceleration of gravity = 32.16, h = head in feet. The actual velocity of discharge in feet per second is V = (l> VJgh = 0.97 Vigh (j) = coefficient of velocity. HYDRAULICS 121 The discharge through the aperture in cubic feet per second is Q = CF<j>\/Ygh = 0,62 pVlJh C = 0.64 (approx.) = coeflftcient of contraction, F = area of aperture in square feet. FLOW OF WATER IN PIPES Bernoulli's Theorem A general method for calculating the flow of water in pipes is given by Bernoulli's theorem* that is, the sum of the velocity head ;r— , the pressure ^ S p head - and the potential head z at any given section of flow is equal to the sum of the corresponding heads at any other section, plus the various losses between the two sections considered. V = velocity in feet per second at first section, I'l = velocity at second section, p = pressure in pounds per square inch at first section, pi = pressure at second section, 122 ENGINEERING MATHEMATICS z = potential head at first section in feet, that is, the distance of the center of the section above a chosen horizontal reference plane, Zi = potential head at second section, g = 32.16 (approx.), 7 = weight in pounds of a column of water 1 foot high and 1 square inch in cross-section = 0.433, k = various losses in feet of head between the two sections of pipe considered. Losses in Pipes The following formulae for losses in pipes enable us to find the value of the term k appearing in Bernoulli's theorem. If several losses occur in a section of pipe, the total loss, kj is the sum of the separate losses. Loss Due to Friction The loss of head in feet due to friction in a sec- tion of pipe is where d = diameter of pipe in feet, I = length of pipe in feet, V = velocity in feet per second, / = coefficient of friction, depending on the velocity, and on the size of pipe. Values of /, the coefficient of friction, for water in clean iron pipes are as follows (condensed from I. P. Church's '^Mechanics of Engineering "): HYDRAULICS 123 Veloc- Diam. 1 :„ Diam. 1 ir-i Diam. Diam. Diam. Diam. Diam. Diam. ity in -2 in. = 2 in. = 4 in. = 8 in. = 12 in. = 16 in. = 20 in. feet per second 0.0417 ft. 0.0834 ft. = 0.1667 ft. = 0.333 ft. =0.667 ft. = 1.00 ft. = 1.333 ft. = 1.667 ft. 0.1 0.0150 0.0119 0.00870 0.00763 0.00704 0.00669 0.00623 0.3 0.0137 0.0113 0.00850 0.00750 0.00693 0.00657 0.00614 0.00578 0.6 0.0124 0.0104 0.00822 0.00732 0.00677 0.00642 0.00603 0.00567 1.0 0.0110 0.00950 0.00790 0.00712 0.00659 0.00624 0.00588 0.00555 2.0 0.00862 0.00810 0.00731 0.00678 0.00624 0.00593 0.00559 0.00529 3.0 0.00753 0.00734 0.00692 0.00650 0.00600 0.00570 0.00538 0.00509 6.0 0.00689 0.00670 0.00640 0.00605 0.00562 0.00534 00507 0.00482 12.0 0.0%30 0.00614 0.00590 0.00560 0.00522 00500 0.00478 0.00457 20.0 0.00615 0.00598 0.00579 0.00549 0.00508 0.00485 Loss at Entrance The loss of head in feet due to entrance from a reservoir into a pipe is equal to in which is the co- \ zJyz:~Ir~~ ^ ^ - eflScient of friction and \—_ ^ -=-> ~ ~~~^ ~ - — is dependent on the \r__rr~_~'/ angle d° which the pipe ^" """ " ^ makes with the inner surface of the reservoir. Values oi Le[ = —^— 1 ) in the above formula are as follows for different values of 6° (from Church) : ^° 90° 80° 70° 60° 50° 40° 30° Le 0.505 0.565 0.635 0.713 0.794 0.870 0.987 Thus, when the discharge is through a pipe normal to the inner surface of the reservoir, then ^° equals 90"^ and Le is, therefore, 0.505, the loss at entrance then being c,2 0.505 2g where v = velocity of flow in pipe in feet per second. 124 ENGINEERING MATHEMATICS Loss Due to Sudden Enlargement The loss of head in feet due to the sudden enlarge- ment of a pipe is .2 (^'J| g Fi = cross-section area of ! ~ the smaller pipe in ; square feet, ^ i F = area of enlarged ^ section in square feet, V = velocity in feet per second in the enlarged section. 2g Loss Due to Sudden Contraction The loss of head in feet due to the sudden contraction of a pipe is in which V = velocity in feet per second in con- "" tracted section, C = coefficient of con- traction, the value of which depends on the F ratio, — , of the small section to the large section. Values of C, the coefficient of contraction, for F HYDRAULICS 125 diflferent values of — are given in the following table (from Church): F_ 0.10 0.624 0.20 0.632 0.30 0.643 0.40 0.659 0.50 0.681 0.60 0.712 0.70 0.755 0.80 0.813 0.90 0.892 1.0 1.0 Loss Due to Bends The loss of head in feet due to a bend in a circular pipe is [o- 131 + 1.847 2g a = radius of pipe in feet, r = radius of bend in feet, V = velocity of flow in feet per second. Values of Lb for different values of - are as follows: r U 0.10 0.131 0.20 0.138 0.30 0.158 0.40 0.206 0.50 0.294 0.60 0.440 0.70 0.661 0.80 0.977 0.90 1.40 1.00 1.98 Flow Through Straight Cylindrical Pipes Q = discharge in cubic feet per second, V = velocity of discharge in feet per second, I = length of pipe in feet, d = diameter of pipe in feet, Le = coefficient of loss at entrance. In general, the pipe is normal to the inner surface of the reservoir and then Le = 0.505. For other cases see Loss at Entrance. / = coefficient of friction, obtained from the table on page 123. 126 ENGINEERING MATHEMATICS (1) Required the head in feet necessary to keep up a given flow of Q cubic feet per second in a clean iron pipe of given length / and diameter d. The required head is /i (in feet) = |^(l + Le + 4/^) 4Q in which v = 7rd^ (2) Required the velocity in the pipe, having given the head h and the length I and the diameter d of the pipe; also required the discharge Q in cubic feet per second. The velocity in feet per second is: .= ■ '^' l + Le+4/^ and after solving for Vy Q = lirdh Since the value of / depends on the unknown v as well as the known J, we may first put / = 0.006 for a trial approximation and solve for v; then take the value of/ corresponding to this velocity and substitute again in the given formula for v. One trial is generally sufficient for ordinary accuracy. (3) Required the proper diameter d for the pipe to discharge a given quantity Q cubic feet per second, having given the length of pipe and the head h. The proper diameter in feet is ^'^'-"^W^c?)' HYDRAULICS 127 and d being solved for, Since the radical contains d, we must first assume a trial value for d, and taking / = 0.006, substitute in the above formula for the diameter. Having obtained a value for d, we solve for the velocity v. With the approximate values of d and v thus obtained, we find the corresponding new value of / from the table of friction, and then substitute again in the formulae. One or two trials generally give sufficient accuracy. Flow Through Very Long Pipes When a pipe is very long (1000 feet or more), the head, velocity, or discharge, etc., may be calculated from the formulae: I 1? /f = 4/- r— (Chezy's formula) dig. 40 Notation same as in preceding section. FLOW THROUGH OPEN CHANNELS Bazin's Formula The velocity of flow in a channel in feet per second is V = 0.552 + -^ Vr 28 ENGINEERING MATHEMATICS r = mean hydraulic radius in feet, which is fou'nd by dividing the area of the fluid cross-section in square feet by the wetted perimeter in feet (that is, the perimeter of the channel section in contact with the water), s = slope of stream (that is, the difference in eleva- tion between two points of the water surface divided by the distance between the two points measured along the surface), m = coefficient of roughness, the values of which are given in the following table. Character of channel Very smooth cement surfaces or planed boards. . Concrete, well-laid brick, unplaned boards Ashlar, good rubble masonry, poor brickwork. . . Earth beds in perfect condition Earth beds in ordinary condition Earth beds in bad condition covered with debris Value of m 0.06 0.16 0.46 0.85 1.30 1.75 Kutter's Formula The velocity of flow in a channel in feet per second equals ,, ^^ , 0.00281 , 1.811 41.65 H 1 _ ,s n /— where r and 5 are as in Bazin's formula. Values for n^ the coefficient of roughness, are as follows: HYDRAULICS 129 Character of channel Planed timber, glazed or enameled surfaces. . Smooth clean cement Unplaned timber, new well-laid brickwork. . . Smooth stonework, ordinary brickwork, iron. Rough ashlar and good rubble masonry Firm gravel Earth in ordinary condition Earth with stones, weeds, etc Earth or gravel in bad condition Value of n 0.009 0.010 0.012 0.013 0.017 0.020 0.025 0.030 0.035 FLOW OVER WEIRS Contraction is complete when no edge of the weir is flush with the sides or bottom of the channel. Contraction is incomplete when one or more sides of the weir have an interior border flush with the sides or bottom of the channel. Francis' Formula The flow over a weir in cubic feet per second is (2 = 1 [0.622 h{h- tV nh) VYgh] in which h = head in feet of water on weir, b = width of weir in feet, n = 2 for complete contraction, n = I ior one end of weir flush with side of channel, n = for both ends of weir flush with sides of channel. Bazin's Formula for Weirs For overfall- weirs with end contractions suppressed, the flow in cubic feet per second is 130 ENGINEERING MATHEMATICS in which the coefficient n has the value 0.0148 n = 0.6075 + h h = depth in feet of water on weir, b = width of weir in feet, p = height in feet of the sill of the weir above the bottom of the channel of approach. STRESSES IN PIPES AND CYLINDERS Pressure in Pipes The tensile stress in pounds per square inch in a pipe due to internal fluid pressure is: fp For thin pipes, P' = ^ t For thick pipes or cylinders, p(r+t) r = inside radius of pipe in inches, t = thickness of pipe in inches, p = excess of internal over external pressure in pounds per square inch. If S is the required factor of safety, then: For thin pipes, t = S-^ For thick pipes or cylinders, rp t P- pS in which r and p are as above, and P = ultimate tensile strength of material of pipe (see Table of Strength of Materials). FLOW OF FLUIDS I31 Collapsing of Tubes The collapsing pressure for Bessemer steel lap- welded tubes, for lengths greater than six diameters, is p =1000 (1 - y 1 - I6O0Q when ^ < 0.023 or p = 86670 3 - 1386 when -. > 0.023 a a (Stewart's equations) in which p = excess of external over internal pressure in pounds per square inch, d = outside diameter of tube in inches, / = thickness of tube wall in inches. FLOW OF FLUIDS Flow of Air Through Apertures The weight of air in pounds discharged per second from a reservoir into the atmosphere is M=0.^3F-|L when pi>2pa or . M = 1.06 F J tAhlJ:A when pi<2pa Fliegner's equations pi = reservoir pressure in pounds per square inch absolute, pa = atmospheric pressure (14.7 pounds per square inch) , F = cross-section of aperture in square inches, T\ = absolute temperature of reservoir (degrees Fahr. + 459-6). 132 ENGINEERING MATHEMATICS Flow of Steam Through Apertures M = 0.0165 Fpi'-^' (Grashof's formula) M = ^ when pi> ^p2 1^ Pp2. I^ipl- P2) T_ . ^ 5 Napier's equations Grashof's formula applies when the final pressure is less than 58 per cent of the reservoir pressure. M = pounds of steam discharged per second, pi = reservoir pressure in pounds per square inch, p2 = final pressure in pounds per square inch, F = cross-section of aperture in square inches. Flow of Gas in Pipes Q = 1000 y^ (Molesworth) Q = quantity of gas in cubic feet per hour, d = diameter of pipe in inches, / = length of pipe in yards, h = pressure in inches of water, 5 = specific gravity of gas relative to air. Flow of Air in Pipes ^ = 114.5 y^ (Hawksley) V = velocity in feet per second, h = head in inches of water, d = diameter of pipe in inches, FLOW OF FLUIDS 133 L = length of pipe in feet, Q = quantity in cubic feet per second. Flow of Compressed Air in Pipes i.0.U61v/^'. 0.1161 v^^' Q = volume in cubic feet per minute of compressed air, at 62° F., Qi = volume before compression, at 62° F., r = pressure in atmospheres, p = difference in pressures in pounds per sq. inch, causing the flow, d = diameter of pipe in inches, L = length of pipe in feet. Flow of Steam in Pipes (Babcock) W = weight of steam flowing in pounds per minute, w = density in pounds per cubic foot of the steam at the entrance to the pipe, pi = pressure in pounds per square inch at the entrance, p2 = pressure at exit, d = diameter in inches, L = length of pipe in feet. 134 ENGINEERING MATHEMATICS ELECTRICITY OHMIC RESISTANCE The resistance of a uniform electric conductor at 0° Centigrade is given by the formula: R (in ohms) = p-j L = length of conductor in inches, A = cross-section in square inches, p = resistivity of conductor at 0° C, values of which are given in the following table. TABLE OF RESISTIVITIES (Resistivity is the resistance in ohms between any two opposite faces of a 1 inch cube of the material)* Metal Resistivity at 0°C. Aluminium (annealed) . . Aluminium (commercial) Aluminium bronze Bismuth (compressed) . . . Brass 1.14 XlO-^ 1.05 XlO-6 4.96 XlO-^ 51.2 XlO-6 2.82 XlO-^ 0.637x10-6 0.625x10-6 8.23 XlO-6 0.803x10-6 3.82 XlO-6 7.68 XlO-6 1.72 XlO-6 37.1 XlO-6 4.89 XlO-6 3.53 XlO-6 0.575x10-6 5.16 XlO-6 2. X10-« 2.28 XlO-6 Copper (drawn) Copper (annealed) German silver . . . Gold (annealed) Iron (wrought) Lead (compressed) Magnesium Mercury Nickel (annealed) Platinum (annealed) Silver (annealed) Tin Tungsten Zinc (pressed) * This definition applies to English units and to the numerical values given in the table. In general, resistivity is the resistance of a unit cube. IS ELECTRICITY 135 The resistance of a conductor at any temperature (1 + ah) in which Ri = known resistance at a temperature h degrees Centigrade, R<2, = required resistance at a temperature h degrees Centigrade, a = temperature coefficient of electrical resistance, the value of which is given for different metals in the following table. TEMPERATURE COEFFICIENTS OF ELECTRI- CAL RESISTANCE Metal Temp, coeffi- cient (approx.) forl°C. Aluminium (commercial) . . Copper (annealed) German silver 0.00435 0.00388 0.00036 0.00365 0.00463 0.00072 0.00247 0.00377 0.00570 0.00365 Gold (annealed) Iron (wrought) Mercury Platinum Silver Tungsten Zinc (pressed) Note. — The temperature coefficient of a material is its increase in resistance for each degree Centigrade rise in temperature, and it is expressed as a decimal fraction of the resistance at 0° C. 136 ENGINEERING MATHEMATICS u Qi .J o — moo >0'>«'0^ 0^\00 0^r<^0 o tr»fso i>.sO»n f<^<N<N — .— »— O (D (DO 00 C — o: r^scsi— sovo-'r <NfNO\ msoo\ r^ •^r^os »— '^oo cr^o^^o vooom <^oooo t^vOi>t tncTiso 000 ,— — — rsi<Nr«^ -^uM^ o^'— -^ oon^os r>.t>»a» odo 000 000 000 o — — ^ — <N<Ni fA'^'m' 0<^0<> OsOON «— f«^0^ VO'^f^ o<Niri om — 000 — — — PvICNfO c<^00— 00 OO-^ 0<NOO 000 000 000 000 O — — ^<N<N o C/3 0^r^— f^OOO vO-^O <^ — -"r — 0<N vTiOvO \0<Nir» <Nin(N — <NvO OsOco ooovo in'«rf<^ <n<n — \0«^0 oo>om -"rr^fN rq — — — oi~ ~ — — — 000 000 000 OOi 000 000 000 000 000 o o o o o o :8: 00c ooc o_^o^c_ >OQ 000 000 000 000 >oo 000 OQr^ f<^00— ifiooiO ir>i— c<>oom — •^<N "^— :;— *^'0.^ n s q '3)0 — 00 vol^ro ^oov oot^i>* vomin m 1 8 s r< » S?§ ^§5: sPiS c^;^^ O — (N rO'^vn \OI>«00 O^O— CNfA^ lAsOr^ ELECTRICITY 137 Is O — Ah a o .s f^ O- &^ oo^ o-'J"'* ■* — — <NOin mor>» r^o^ fsoov m-Tf-^t- <Nt^so — Qooo o-fo tncNo t>»^^ 52'^ o^o^o "^o^io cNo^r^ vO'^f<^ <^rvj— — — o oco oo 3 000 000 000 00 CN — — —00 T^o^ —00 CN — O vO-^cr* 173 r>N oil O vO 00 o — tnoO CNOvO >A<N'^ — vom fOfNf*^ 000 <N<Nr<^ -"rinvo 000 03 00 — <NOQO sii moo —mo mfs vOm-rr rO<N«N — — §88 888 88 000 000 000 000 000 000 000 ■^«N00 sOvOr<^ 0000 ON — ■«»■ OMncN o^rs. c<^f<^r^^ _,— — 00 o 000 00 - 000 00 888 88 000 00 000 <NOv<N vO«NO 2i;!8 oovcm -n-focN cs — — (NOI^ Tfo otNin ovOtN scm -nr 2. ^ P5 00f<^0 lAOvO — OOs <NvOcO OC^O^ _c<^so <^ c*^ c<^ ^iQ« ?:|^Si 00 NO in lATfcn CM c o O <u |8 inovo — 0>0n (NvOcn OONO — <r\vo in f<^«S<N ;5;?52: oosom •<^<N — — oo^ 00 in (^ vo — o^On p^ vO r<^ oo^o 1 0000 — mso t>ivOin in — men ooo^o — p^m '♦•invo r>.ooa^ o — <n pf^'^m — — <N <N«Nfs <NPvjp^ <N<NCN m co m cn m m OnO 138 ENGINEERING MATHEMATICS Ohm's Law or E = IR I = current in amperes, E = electromotive force in volts, R = resistance in ohms. The proper size of wire in circular mils for any direct current circuit on a two-wire system consisting of copper conductors is given by the formula: 10.8 X 2dXl cm. = — -g - or if the resistance is required, E r = 2dXl where r = resistance per foot of wire in ohms, E = volts drop in line, / = total line current in amperes, d = distance from source to load in feet, cm. = cross-section of conductor in circular mils. Resistance of Circuits The resultant of several resistances in series equals R = ri + r2 + r3+ • • • where ri, ^2, ^3, etc., are the separate resistances. The resultant of several resistances in parallel or multiple is given by the relation: w • 1+1+1+.. . R n r2 Tz ELECTRICITY 139 R is the total or combined resistance; and ri, ^2, rs, etc., are the separate resistances. Power and Energy in Direct Current Circuits The power in watts expended in a resistance is P = EI = PR E = electromotive force in volts, / = current in amperes, R = resistance in ohms. The energy transformed into heat in a time t seconds is e = EIt= PRt when the current, /, is constant; or, if the current is variable, energy equals !> i^Rdt h where i is the instantaneous value of the current, expressed as a function of t. The power in any two-wire direct current circuit is P (in watts) = EI where E is the volts between the terminals of the cir- cuit and / is the current in amperes. MOTORS AND GENERATORS The frequency in cycles per second is given by the relation: _ R.P.M. P •^ " 60 ^2 R.P.M. = speed in revolutions per minute, P = number of poles. I40 ENGINEERING MATHEMATICS Equations of Direct Current Motor The armature current of a motor, during starting, is Ra H~ Rx in which E = impressed voltage, e = counter-electromotive force, Ra = armature resistance in ohms, Rx — resistance of grid or rheostat in series with armature. At full speed, E- e ^"" Ra e = K(j>f E =IaJla+e=IaRa+K<t>f E - K(j>f Ra E — IgRa K<t> f = frequency in cycles per second,* </) = total field flux in magnetic lines, cutting arma- ture conductors, K = constant for any given machine. Its value is 4/ 77^> where t is the number of armature turns in series. * Frequency, in the case of a direct current machine, refers to the frequency of alternation in the armature windings, not, of course, in the external circuit* ELECTRICITY 141 Equations of Direct Current Generator E = e- laRa e = generated voltage, E = terminal voltage, la = armature current in amperes, Ra = armature resistance in ohms. R = resistance of load in ohms. E^RIa e=E+IaJla=Ia(R+Ra) Torque The torque of a dynamo in foot-pounds equals T = KI<I> where <^ = total field flux in magnetic lines, cutting arma- ture conductors, / = armature current in amperes, K = constant term for any given dynamo. Its value 2.348 is K = ' Q tP, t being the number of arma- ture turns in series, and P the total number of poles. The torque of a motor in terms of the horsepower is ^^ 33,000 H.P. or solving for horsepower, lirTn IwRFn H.P. = 33,000 33,000 142 ENGINEERING MATHEMATICS n = number of revolutions per minute, T = torque in foot-pounds, R = radius of pulley in feet, F = turning force in pounds. Induced Voltage N dci> . ' = ^WTt ^^^^' N = number of turns. If the turns cut across a uniform field, at right angles to the lines of force, then -j- equals the number of lines cut per second. Otherwise, — is the first derivative of ^ in respect to t, (j> being expressed as a function of /. The efifective voltage induced in the windings of a generator, motor, or transformer, etc., is given by the relation: ^ V2Tfn(t> 4.44/^0 ,, £ = -I^=--li- volts This formula is generally quite accurate, being derived on the assumption of uniform flux distribution. / = frequency in cycles per second, (l> = total number of lines of magnetic force, n = effective number of turns. If all the turns are grouped in one coil, then n equals the total number of turns. Otherwise, if the winding is distributed over k electrical degrees (as in the armature of a motor or generator), then the effective number of turns isn = N k 2 N being the total number of turns. i ELECTRICITY 143 The average induced voltage of a dynamo is E - 4^ volts where n is the number of armature turns in series. Inductance Inductance, L, is the number of interHnkages of flux with turns, per unit current, L (henrys) = j^ in which N = number of turns, / = current in amperes, (j> = number of Hues of magnetic force interhnking with the turns. The theoretical unit of inductance is the centimeter. The practical unit of inductance is the henry, which equals 10^ centimeters. The counter-electromotive force in an inductive circuit is ^ di provided the inductance, L, is constant. The total voltage consumed by an inductive circuit E.ir+L% the inductance, L, being constant. di . r is the resistance of the circuit in ohms, and -r- is the first derivative of i with respect to ^, the current i being expressed as a function of t. 144 ENGINEERING MATHEMATICS The inductance in henrys of an air-core circular coil is 0.366 ■ L = Viooo/ F' F 106+ 12c + 2Jg 106+ 10c+ lAR UR XF'F" " = 0.51ogio( 100 + 2 & + 3 c/ I = length of conductor in feet. — - -T 1 - f a r I 1 • b 1- t 1 All other dimensions are in inches and as indicated in the diagram. The inductance, L, of a concentric cable in henrys per 1000 feet is L - 105 X j^+4.61og.o^+^^^,_^,^, 4.6 J?o^ ^0 l 3Ro^- R^ l where r = radius of inner metallic conductor, R = distance from center of cable to the inner sur- face of the outer metallic conductor, Ro= distance from center of cable to the outer sur- face of the outer metallic conductor. ELECTRICITY 145 The values of r, R, and Rq must be expressed in the same units. The total inductance, L, of a two- wire transmis- sion circuit in henry s per 1000 feet is ^ 3.048 (^^ , D-r , where /zi = permeabihty of the metal conductor; for copper, Ml = 1, IX = permeability of medium separating wires; for air, jjL= ly D = distance between the two lines, measured from center to center, r = radius of conductor, in same unit as D. Capacity The unit of capacity is the farad. Since the farad is very large, the microfarad, which is one-millionth of a farad, is used as the practical unit. The theoretical unit of capacity is the centimeter, 9 X 10^^ centimeters being equal to 1 farad. The charge of a condenser, Q, is measured in ampere- seconds or coulombs, and may be calculated by the formula: Q = CE from which ^=1 and where C = capacity in farads, E = potential across the terminals of the condenser m volts. 146 ENGINEERING MATHEMATICS The capacity of a plate condenser is 22AS KA C = microfarads I d X W where A = total area in square inches of all the dielectric sheets separating the condenser plates, d = average thickness in inches of one sheet of the dielectric, K = inductivity of the dielectric, average values of which are given in the following table for different materials. Materials Induc- tivity K Air (at standard pressure). Manilla paper 1.00 1.50 2.00 2.50 2.50 3.00 3.00 3.10 6.00 Paraffin, solid Ebonite India rubber SheUac Oil Glass Mica Condensers in Parallel. When two or more con- densers are connected in parallel, the resultant capacity, C, equals the sum of the separate capacities, thus C=Ci+C2 + C3+ .... Condensers in Series. When two or more con- densers of capacities Ci, C2, C3, etc., are connected in series, the resultant capacity is given by the formula: 1 C = -^ + ^- + ^ + ELECTRICITY 147 The capacity, C, of a concentric cable per 1000 feet in microfarads is 7.37 C = lOOOlogio- P in which p = radius of inner metallic conductor, Po = distance from center of cable to the inner sur- face of the outer metallic conductor, in the same unit as p. The capacity, C, of a two-wire transmission line per 1000 feet in microfarads is given approximately by the formula: 3.68 C = lOOOlogio^ i r if the lines are not close to the ground. D = distance between the two wires of the trans- mission line, measured from center to center, r = radius of conductor, in same unit as D. The differential equations of a condenser are dg = idt q = charge = j idt dq = cde de Alternating Current Circuits The shape of the voltage or current wave produced by an alternator is, in general, nearly that of a sine curve. Alternating current calculations are, therefore, usually worked out on this assumption. 148 ENGINEERING MATHEMATICS The number of cycles or complete waves per second is the frequency of the current, and the time required for the current to complete one cycle is a period. The average value of the current or voltage is the average of all the ordinates of the curve of one half- wave. The effective value of an alternating current or voltage is the square root of the sum of the squares of the instantaneous values of a half-wave. If E is the maximum voltage of a half-cycle of a sine wave, 2 average voltage =-£ = 0.636^ TT 1 eflfective voltage = —rzE = 0.707 E Similarly, if the maximum current is /, 2 average current = -/ = 0.636/ TT 1 effective current = 77^/ = 0.707/ When the voltage reaches a definite value in the cycle sooner than the current reaches its corresponding value, the voltage and current are out of phase with each other; the voltage is said to be leading, and the current to be lagging. Phase difference is always expressed in degrees; a complete cycle equals 360 degrees. ELECTRICITY 149 Alternating Voltage and Current or E = IZ I = current in amperes, E = electromotive force in volts, Z = impedance in ohms. Impedance and Reactance r = resistance in ohms X = reactance in ohms z = impedance in ohms The relation between resistance, reactance, and im- pedance is the same as that between the three sides of a right triangle. r = z cos a X = zsina a = tan ^- r z = VrM^ Inductive Circuits The inductive reactance in ohms is Xl = 27r/L where / = frequency in cycles per second, L = inductance in henrys. The impedance in ohms is z = VrM^ = V/'2+47r2/2L2 150 ENGINEERING MATHEMATICS Circuits having Capacity The capacity reactance in ohms is 1 ItJC where / = frequency in cycles per second, C = capacity in farads. The impedance in ohms is Circuits having Inductance and Capacity The reactance in ohms is X = xl + xc= 2 tt/L - j^ The impedance in ohms equals z = Vr^+{xL + xcY Vector Representation of Sine Waves A sine wave of voltage or current may be represented by a vector, the magnitude or length of which is equal to the effective value of the sine wave. It is some- times more convenient to let the length of the vector equal the maximum value of the sine wave. The vector is generally denoted by a capital letter, with a dot directly beneath it; it is expressed in terms of its rectangular components, which determine the magni- tude of the vector and its direction relative to the coordinate axes. Thus, the vector E is written E = e + je' in which j = V— 1 where e denotes the horizontal or real component of the ELECTRICITY 151 vector, and e' the vertical or imaginary component. The imaginary unit, 7, in the above equation, merely denotes the direction of measurement of e\ The magnitude of E is E = Ve^ + e'2 and the angle B which the vector E makes with the horizontal axis is d tan-i - The angle in degrees between two vectors is the phase difference between the two sine waves which the vectors represent. In vector notation, the impedance is Z =^ r -\- ]x and its magnitude is Z = Vr2 + x^ The admittance is Z r -{- jx r . X , where g = ^ = conductance, * = -| susceptance. 152 The ENGINEERING MATHEMATICS current equals E r = ~ = EY = {e+je') + ji' and the voltage is E = IZ = (i+ji') {r +jx) = e +jV Power in Alternating Current Circuits If the effective voltage and current are represented by the vectors E = e+jV E A T / ^^ • ^ y^6 ^ e' • f — e "T k the real power is IF = a + eV = EI cose the wattless power is Wi = e'i- ei' = EI sinff the volt-amperes equals EI, The power-factor is the cosine of the angle between the voltage and current vectors, power-factor = cos ^ = — ^ — ELECTRICITY 153 Balanced Three-phase Circuits E = volts between lines e = volts per phase / = current in each line i = current in each phase Line Phase Y connection For Y-connections, E = e Vs ; e = —7= ; and I =i V3 Line ;Phase ^ - connection For A-connections, £ = ^; / = i V3; and i = —tz^ V3 In either case, for non-inductive load, the power in watts is W = V3EI 154 ENGINEERING MATHEMATICS If the load is inductive, then the power is W = V3 EI cosd where cos 6 is the power-factor of the phase. MAGNETISM Equations of Magnetic Circuits F = attractive or repellent force in dynes, mmf = magnetomotive force in ampere turns, N = number of turns, / = current in amperes, jS = density in magnetic lines per square centi- meter, <j> = total number of lines of flux, A = cross-section of magnetic path in square centimeters, fjL = permeability, H = intensity of field, I = length of magnetic circuit in centimeters, p = reluctance, m = pole strength, r = distance between poles. QAttNT V ■ p = 4> : 0.4 P ttNIhA I ^■ . A mmf 0.4 = 0.4 tNIh I irNI M H MAGNETISM ISS Magnets and Magnetic Fields F = mH I ^ mm (f) = Arirm The attractive force in pounds exerted by a two pole magnet is P = y ^^^ , where 5 is the total area of both pole faces in square inches, and B is the density in magnetic lines per square inch. The ampere-turns required to maintain a flux den- sity of B lines per square inch in an air gap is IN = 0.313 Blj in which I is the length of the gap in inches. Hysteresis Loss The power in watts lost in hysteresis is , fVB'-' W 107 / = frequency in cycles per second, V = volume of iron in cubic inches, B = magnetic density in lines per square inch, k = empirical constant, values of which are given in the following table. Character of iron Value of k Silicon steel (J. 0006 to 0.00075 0.0008 to 0.0011 0.010 to 0.012 0.013 to 0.017 Annealed sheet iron. . . . Cast steel. . . Cast iron 156 ENGINEERING MATHEMATICS Eddy Current Loss The power in watts lost due to eddy currents in iron or steel laminations is approximately / = frequency in cycles per second, I = average thickness of lamination in inches, B = magnetic density in lines per square inch, V = volume of iron in cubic inches. This formula holds for ordinary temperatures, and if the thickness of the lamination is not greater than 0.025 inch. In silicon steel, the eddy current loss is approximately \ of that given above. STANDARD SATURATION CURVES B = density in lines per square inch A T/in. = ampere-turns per inch Values of ampere-turns per inch for densities not included in the following tables may be determined approximately by interpolation. Thus, the AT/in. for silicon steel for B/sq.in. = 65,500 is AT/in, = 4.5 + ^^ (6.4 - 4.5) = 5.5 (approx.) MAGNETISM 157 Silicon Steel Annealed Sheet Iron Saturation curve Saturation curve B 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 110,000 AT /in. 2.1 2.7 3.4 4.5 6.4 10 23 35 100 225 520 1000 2200 3770 5330 6900 B 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 AT /in. 4 4.4 5 9 12 20 33 60 120,000 130,000 135,000 140,000 145,000 150,000 155,000 Cast Steel Cast Iron Saturation curve Saturation curve B 50,000 60,000 70,000 I 80,000 f 90,000 100,000 105,000 AT /in. 11 15 20 29.5 50 105 165 B 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000 AT /in. 8 12 17 23 30 43 60 85 110 145 190 158 ENGINEERING MATHEMATICS MEASUREMENT English Weights and Measures Length 1000 mils = 1 inch , 12 inches = 1 foot 3 feet = 1 yard 5280 feet = 1 mile 4 inches = 1 hand 9 inches = 1 span 2 J feet = 1 pace 16| feet or 5^ yards = 1 rod 1 knot or nautical mile = 6080.26 feet = I league 7.92 inches = 1 link 25 links = 1 rod 100 links or 66 feet or 4 rods = 1 chain 10 chains = 1 furlong 8 furlongs = 1 mile Surface 144 square inches = 1 square foot 9 square feet = 1 square yard 30i square yards = 1 square rod 160 square rods = 1 acre 640 acres = 1 square mile 625 square links = 1 square rod 16 square rods = 1 square chain 10 square chains = 1 acre 640 acres = 1 square mile 36 square miles = 1 township Volume 1728 cubic inches = 1 cubic foot 27 cubic feet = 1 cubic yard 128 cubic feet = 1 cord 24f cubic feet = 1 perch MEASUREMENT 159 Troy Weight 24 grains (gr.) = 1 pennyweight (dwt.) 20 pennyweights = 1 ounce (oz.) 12 ounces = 1 pound (lb.) Avoirdupois Weight 16 drams (dr.) = 1 ounce (oz.) 16 ounces = 1 pound (lb.) 25 pounds = 1 quarter (qr.) 4 quarters = 1 hundred weight (cwt.) 20 hundred weight (2000 pounds) = 1 ton (T.) Apothecaries' Weight 20 grains (gr.) = 1 scruple (sc. or 9) 3 scruples = 1 dram (dr. or 3) 8 drams = 1 ounce (oz. or §) 12 ounces = 1 pound (lb) Dry Measure 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) 36 bushels = 1 chaldron (ch.) Liquid Measure 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 31i gallons = 1 barrel (bar.) 63 gallons = 1 hogshead (hhd.) Apothecaries' Fluid Measure 60 minims = 1 fluid-drachm 8 fluid-drachms = 1 fluid-ounce 16 fluid-ounces = 1 pint 8 pints = 1 gallon Circular Measure 60 seconds (") = 1 minute (') 60 minutes = 1 degree (°) 30 degrees = 1 sign (s) 12 signs, or 360 degrees = 1 circle (cir.) i6o ENGINEERING MATHEMATICS English and Metric Conversion Factors 1 millimeter 1 centimeter 1 inch 1 foot 1 yard 1 meter 1 kilometer 1 mile Length = 39.37 mils = 0.03937 inch = 0.3937 inch = 0.0328 foot = 2.54 centimeters = 0.083 foot = 30.48 centimeters = 0.305 meter = 91.44 centimeters = 0.914 meter = 39.37 inches = 3.28 feet = 1.094 yards = 3280.8 feet = 1093.6 yards = 0.6214 mile = 5280 feet = 1609.3 meters = 1.609 kilometer 1 circular mil 1 square mil 1 sq. millimeter 1 sq. centimeter 1 sq. inch 1 sq. foot Surface = 0.7854 square mil = 0.0005067 square millimeter = 1.273 circular mils = 0.000645 square miUimeter = 0.000001 square inch = 1973 circular mils = 1550 sq. mils = 0.00155 sq. inch = 197,300 circular mils = 0.155 sq. inch = 1,273,240 circular mils = 6.4516 sq. centimeters = 929.03 sq. centimeters = 144 sq. inches MEASUREMENT i6i 1 sq. yard 1 sq. meter 1 are 1 acre 1 hectare 1 sq. kilometer 1 sq. mile 1296 sq. inches 9 sq. feet 0.00836 are 0.000207 acre 1550 sq. inches 10.76 sq. feet 1.196 sq. yards 1076 sq. feet 100 sq. meters 43,560 sq. feet 4840 sq. yards 4047 sq. meters 0.4047 hectare 0.001562 sq. mile 107,600 sq. feet 100 ares 2.471 acres 10,764,111 sq. feet 247 acres 0.3861 sq. mile 27,878,400 sq. feet 640 acres 2.59 sq. kilometers Volume 1 cu. centimeter = 0.061 cu. inch = 0.0021 pint (liquid) = 0.0018 pint (dry) 1 cu. inch = 16.39 cu. centimeters = 0.0173 quart (hquid) = 0.01488 quart (dry) = 0.0164 liter or cu. decimeter = 0.004329 gaUon = 0.0005787 cu. foot 1 quart (liquid) = 2 pints (liquid) = 946.36 cu. centimeters = 57.75 cu. inches = 0.94636 liter or cu. decimeter 1 quart (dry) = 2 pints (dry) l62 ENGINEERING MATHEMATICS 1 quart (dry) 1 liter 1 gallon = 1101 cu. centimeters = 67.20 cu. inches = 0.03889 cu. foot = 1000 cu. centimeters = 61.023 cu. inches = 1.0567 quarts (dry) = 0.2642 gallon = 3785 cu. centimeters = 231 cu. inches = 3.785 liters = 0.1337 cu. foot Note. Pints, quarts, and gallons in this table refer to U.S. measures. 1 milligram 1 grain 1 gram 1 ounce (av.) 1 poimd (av.) 1 kilogram 1 ton (short) 1 ton (metric) Weight = 0.01543 grain = 0.001 gram = 64.80 milligrams - 0.002286 ounce (av.) = 15.43 grains = 0.03527 ounce (av.) = 0.002205 pound (av.) = 437.5 grains = 28.35 grams = 0.0625 pound (av.) = 7000 grains = 453.6 grams = 16 ounces = 0.4536 kilogram = 35.27 ounces = 2.205 pounds = 2000 pounds (av.) = 907.2 kilograms = 0.8928 ton (long) = 0.9072 ton (metric) = 2205 pounds = 1000 kilograms = 1.102 ton (short) • = 0.9842 ton (long) MEASUREMENT 163 1 ton (long) = 2240 pounds = 1.12 ton (short) = 1.016 ton (metric) Force 1 dyne = 0.01574 grain = 0.00102 gram = 0.00007233 poundal = 0.000002248 pound (av.) 1 gram = 980.6 dynes = 0.07093 poundal 1 poimdal = 13,825 dynes = 0.03108 pound = 0.01410 kilogram 1 pound = 444,800 dynes = 32.17 poundals 1 kilogram = 980600 dynes = 70.93 poundals Storage of Water 1 acre-foot = 325,800 gallons = 43,560 cu. feet = 1613 cu. yards = 1233 cu. meters 1 gallon = 0.000003069 acre-foot 1 cu. foot = 0.00002298 acre-foot 1 cu. yard = 0.00062 acre-foot Temperature 1 degree Centigrade = | (= 1.8) degree Fahrenheit 1 degree Fahrenheit = f ( = 0.556) degree Centigrade temperature Fahr. = // = | /c + 32 temperature Cent. = ^c = I (^/ — 32) 164 ENGINEERING MATHEMATICS 1 gram-centimeter 1 joule 1 foot-pound Heat, Electric, and Mechanical Equivalents Energy 1 erg = 1 dyne-cm. = 0.0000001 joule = 0.00000007376 foot-pound = 980.6 ergs = 0.00009806 joule = 0.00007233 foot-pound = 10,000,000 ergs = 0.7376 foot-pound = 0.2389 gram-calorie = 0.102 kilogram-meter = 0.0009480 B.t.u. = 0.0002778 watt-hour = 13,560,000 ergs = 1.356 joules = 0.3239 gram-calorie = 0.1383 kilogram-meter = 0.001285 B.t.u. = 0.0003766 watt-hour = 0.0000005051 horsepower-hour = 9.806 joules = 7.233 foot-pounds = 0.009296 B.t.u. = 0.002724 watt-hour = 1055 joules = 778.1 foot-pounds = 252 gram-calories = 107.6 kilogram-meters = 0.2930 watt-hour = 0.0003930 horsepower-hour = 3600 joules = 2655.4 foot-pounds = 860 gram-calories = 3.413 B.t.u. = 0.001341 horsepower-hour = 4186 joules = 3088 foot-pounds 1 kilogram-meter 1 B.t.u. 1 watt-hour 1 kilogram-calorie MEASUREMENT 165 1 kilogram-calorie = 426.9 kilogram-meters = 1.163 watt-hours 1 horsepower-hour = 2,684,000 joules = 1,980,000 foot-pounds = 745.6 watt-hours Power 1 erg per second = 1 dyne-centimeter per second = 0.0000001 watt 1 gram-centimeter per second = 0.00009806 watt 1 foot-pound per minute = 0.02260 watt = 0.00003072 horsepower (metric) = 0.00003030 horsepower 1 watt = 44.26 foot-pounds per minute = 6.1 19 kilogram-meters per minute 1 horsepower = 33,000 foot-pounds per minute = 745.6 watts = 550 foot-pounds per second = 1.01387 horsepowei (metric) 1 horsepower (metric) = 32,550 foot-pounds per minute = 735.5 watts = 75 kilogram-meters per second = 0.9863 horsepower 1 kilowatt = 44,256.7 foot-pounds per minute = 1.3597 horsepower (metric) = 1.341 horsepower Electric Units 1 abvolt = 10-8 volt 1 abampere = 10 amperes 1 abohm = 10-® ohm Pressure Eqiiivalents B 1 atmosphere (standard) = 29.92 12 inches of mercury at 32° F. = 760 miUime ters of mercury at 32 ° F. = 33.901 feet of water at 39.1° F. = 14.6969 pounds per sq. inch = 2116.35 pounds per sq. foot ( l66 ENGINEERING MATHEMATICS 1 inch of mercury at 32° F. = 0.491187 pound per sq. inch = 70.7310 pounds per sq. foot = 1.13299 feet of water at 39.1° F. 1 foot of water at 39.1° F. = 0.8826 inch of mercury at 32° F. = 62.425 pounds per sq. foot = 0.4335 pound per sq. inch = 0.0295 atmosphere 1 pound on the sq. foot = 0.016018 foot of water at 39.1° F. 1 pound on the sq. inch = 2.307 feet of water at 39.1° F. PRESSURE AND VOLUME CORRECTION, ETC. Reduction of Barometer Readings to o"" C, (/5 - a) t] corrected height Ho = H\l — (i + m H = observed height of barometer, / = observed temperature of barometer in degrees Centigrade, j8 = 0.0001818, the coefficient of cubical expansion of mercury, a = coefl&cient of linear expansion of the material of the scale (0.0000085 for glass, 0.0000184 for brass). Reduction of Gaseous Volumes to o° C, and I Atmosphere Pressure corrected volume Vo -k + 0.00367/760 V = observed volume, / = observed temperature in degrees Centigrade, p = pressure in millimeters of mercury. MEASUREMENT 167 Determination of Altitudes by the Barometer For heights not exceeding 2000 feet, relative altitude is given by the approximate formula: 2 (r + Ti)] H-Hi Z (in feet) = 52,500 1 + 1000 H + Hi X = vertical distance between the two stations, T = Centigrade temperature at lower station, Ti = Centigrade temperature at upper station, H = height of barometer at lower station reduced to 0° C, Hi = height of barometer at upper station reduced to 0° C. For any altitude, Z = 60,346ll+0.00256cos(2^) Jl + ^^^^Wgio^ in which d = latitude in degrees. i68 ENGINEERING MATHEMATICS PHYSICAL AND CHEMICAL CONSTANTS ATOMIC WEIGHTS Element Aluminium . Antimony. . Argon Arsenic Barium Beryllium. . Bismuth. . . . Boron Bromine. . . . Cadmium. . . Caesium .... Calcium. . . . Carbon Cerium Chlorine. . . . Chromium. . Cobalt Copper Dysprosium. Erbium Europium. . . , Fluorine Gadolinium. Gallium Germanium. Gold Helium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum. . Lead Lithium Lutecium. . . . Magnesium. . Manganese. . Mercury Molybdenum Sym- Atomic bol weight Al 27.1 Sb 120.2 A 39.88 As 74.96 Ba 137.37 Be 9.1 Bi 208.0 B 11.0 Br 79.92 Cd 112.40 Cs 132.81 Ca 40.07 C 12.00 Ce 140.25 CI 35.46 Cr 52.0 Co 58.97 Cu 63.57 Dy 162.5 Er 167.7 Eu 152.0 F 19.0 Gd 157.3 Ga 69.9 Ge 72.5 Au 197.2 He 3.99 H 1.008 In 114.8 I 126.92 Ir 193.1 Fe 55.84 Kr 82.9 La 139.0 Pb 207.10 Li 6.94 Lu 174.0 Mg 24.32 Mn 54.93 Hg 200.6 Mo 96.0 Element Neodymium. . Neon Nickel . Niobium Nitrogen Osmium Oxygen Palladium .... Phosphorous. . Platinum Potassium. . . . Praseodymium Radium Rhodium Rubidium .... Ruthenium. . . Samarium. . . . Scandium Selenium Silicon Silver Sodium Strontium. . . . Sulphur Tantalum Tellurium Terbium Thallium Thorium. ..... Thulium Tin Titanium Tungsten Uranium Vanadium. . . . Xenon Ytterbium. . . . Yttrium Zinc Zirconium. . , Sym- bol Nd Ne Ni Nb N Os O Pd P Pt K Pr Ra Rh Rb Ru Sa Sc Se Si Ag Na Sr S Ta Te Tb Tl Th Tm Sn Ti W U V Xe Yb Y Zn Zr Atomic weight 144.3 20.2 58.68 93.5 14.01 190.9 16.00 106.7 31.04 195.2 39.10 140.6 226.4 102.9 85.45 101.7 150.4 44.1 79.2 28.3 107.88 23.00 87.63 32.07 181.5 127.5 159.2 204.0 232.4 168.5 119.0 48.1 184.0 238.5 51.06 130.2 172.0 89.0 65.37 90.6 PHYSICAL AND CHEMICAL CONSTANTS 169 WEIGHT AND DENSITY OF VARIOUS ._ SUBSTANCES Values are for ordinary temperatures unless otherwise stated. Metals Aluminium Antimony Bismuth Brass (ordinary) Bronze Calcium Copper (pure) Copper (cast) 39° F.. Copper (rolled) 39° F Gold Iron (pure) Iron (cast) 39° F Iron (wrought) 39° F Lead Magnesium Mercury 32° F Nickel Platinum Potassium Silver Sodium Steel (hard) 39° F.. . Steel (soft) 39° F Tin Tungsten Zinc Weight in pounds per cu. in. per cu. ft. 0.096 166.5 0.244 422.0 0.354 612.0 0.308 532.0 0.319 552.0 0.057 98.5 0.322 565.0 0.314 541.5 0.321 554.0 0.698 1205.0 0.284 490.0 0.260 449.0 0.281 485.0 0.411 709.7 0.064 109.0 0.491 848.0 0.318 549.0 0.775 1340.0 0.031 53.9 0.379 655.0 0.035 60.5 0.286 494.0 0.283 488.0 0.264 455.0 0.624 1080.0 0.253 437.0 Density relative to water 2.67 6.76 9.82 8.55 8.85 1.58 8.93 8.70 8.88 19.32 7.86 7.21 7.78 11.38 1.75 13.60 8.80 21.50 0.87 10.50 0.97 7.92 7.83 7.30 17.30 7.00 170 ENGINEERING MATHEMATICS WEIGHT AND DENSITY OF VARIOUS SUBSTANCES (Continued) Liquids Weight in pounds Density relative to water Acid, hydrochloric per cu. in. 0.0433 0.0440 0.0675 0.0286 0.0455 0.0455 0.0307 0.0332 0.0328 0.0307 0.0314 0.036121 0.036125 0.036085 0.034549 0.037023 per cu. ft. 74.8 76.0 116.5 49.5 78.5 78.5 53.0 57.4 56.6 53.0 54.1 62.417 62.425 62.355 59.700 63.976 1 20 Acid, nitric 1.22 Acid, sulphuric 1.84 Alcohol 0.79 Carbon disulphide 1.26 Glycerine 1.26 Naphtha. . . . 85 Oil,, linseed 0.92 Oil, lubricating 0.91 Petroleum 0.85 Turpentine. ... 87 Water, pure, at 32° F. (freezing point) . . . 39.1° F. (max. density)... 62° F. (standard temp.). . 212° F. (boiling point)... . Water, Sea, 62° F 1.0010 1.0011 1.0000 0.9574 1.0260 Values for gases given below are for 32° F. and a pressure of i atmosphere. Gases Acetylene, C2H2 Air Ammonia, NH3 Carbon monoxide, CO. . Carbon dioxide, CO2. . . Ethylene, C2H4 Hydrochloric acid, HCl. Hydrogen, H2 Hydrogen sulphide, H2S Methane, CH4 Nitrous oxide, N2O. . . . Nitric oxide, NO Nitrogen, N2 Oxygen, O2 Sulphur dioxide, SO2. . . Water vapor, H2O Weight in pounds per cu. ft. 0.0725 0.0807 0.0475 0.0781 0.1227 0.0781 0.1023 0.00562 0.0949 0.0446 0.1235 0.0831 0.0783 0.0892 0.1786 0.0502 Density relative to 0.898 1.000 0.589 0.967 1.520 0.967 1.268 0.0695 1.175 0.553 1.530 1.030 0.970 1.105 2.210 0.622 ^ PHYSICAL AND CHEMICAL CONSTANTS 171 WEIGHT AND DENSITY OF VARIOUS SUBSTANCES {Continued) Woods Ash Beech Cedar , Cork Elm Fir Lignum-litae , Mahogany. . , Maple Oak Pine, Yellow- Pine, White. 'Poplar Spruce Walnut Weight in in pounds per cu. ft. 45 46 39 15 38 37 62 51 42 47 38 28 30 28 36 Density relative to water 0.72 0.73 0.62 0.24 0.61 0.59 1.00 0.81 0.68 0.75 0.61 0.45 0.48 0.45 0.58 Other materials Asphaltum Brick, common. . . Cement, average.. Clay Coal, anthracite. . Coal, bituminous. Concrete, average Earth, loose Earth, packed. . . . Glass, average. . . . Glass, flint Granite Gravel, average. . . Ice Limestone Marble Quartz Sand, average Slate Weight in pounds percu. ft. 87 112 90 135 95 84 135 75 100 164 188 165 110 56 165 170 165 100 175 Density relative to water 1.39 1.79 1.45 2.15 1.50 1.35 2.20 1.20 .60 .60 .02 ,65 1.75 0.90 2.65 2.73 2.65 1.60 2.80 172 ENGINEERING MATHEMATICS MELTING AND BOILING POINTS OF ELEMENTS Element Melting point Boiling point at atmospheric pressure Degrees C. Degrees F. Degrees 1800 1440 -186 / subl \450 1420 ' / subl \3500 63 778 -33^6 2200 2310 ' -187 2530 -268.6 -252.7 184.4 2450 1525 1400 1120 1900 356.7 2330 -195.7 -182*9 2540 287 2450 758 690 3500 1955 877 Degrees Aluminium Antimony Areon 657 630 -188 1214 1166 -306 3272 2624 -303 Arsenic (volatilizes") imes \ 842 1 Barium 850 269 2000 -7.3 321 780 4000 -102 1489 1490 1083 -223 1062 below -2 70 -259 113 1505 327 186 633 1207 -38.8 1452 -210.5 2200 -235 1549 44.1 1710 62.5 217 1420 960 97.0 1562 516 3630 18.8 610 1436 7230 -151.5 2712 2714 1982 -370 1944 below— 454 -434 235 2742 621 367 1172 2205 -37.8 2648 -347 3990 -391 2820 111.5 3110 144.5 423 2588 1760 206.6 Bismuth 2590 Boron imes \ 6330 I 145.5 Bromine Cadmium 1432 Calcium Carbon Chlorine -28.5 Chromium Cobalt 3992 Copper 4190 Fluorine -305 Gold 4586 Helium -452 Hydrogen Iodine -423 364 Iron 4442 Lead 2776 Lithium 2552 Magnesium Manganese Mercury 2052 3452 674 Nickel 4226 Nitrogen -320 Osmium Oxvs^en -297 Palladium Phosphorous Platinum 4600 549 4440 Potassium Selenium 1397 1274 Silicon 6330 Silver 3551 Sodium 1612 ^ PHYSICAL AND CHEMICAL CONSTANTS 1 73 MELTING AND BOILING POINTS OF ELEMENTS {Continued) Element Melting point Boiling point at atmospheric pressure ^ Degrees C. . Degrees F. Degrees C. Degrees Strontium Sulphur (rhombic) Tantalum Tin 900 115 2910 232 2500 3083 418 1300 1650 239 5270 449.6 4530 5582 784 2372 '445' 2270' 3700" 918 '833' 4122 Titanium Tungsten 6700 Zinc 1683 Zirconium SPECIFIC HEATS The values of specific heat, unless otherwise stated, are average values, and hold approximately over ordinary ranges of tempera- tures. Solids Aluminium . . . . Antimony Bismuth Brass Copper Gold Iodine Iron (wrought) Iron (cast) . . . . Lead Magnesium . . . . Manganese. . . . Nickel Phosphorous. . . Platinum Silicon Silver Steel Sulphur Tin Tungsten Zinc Specific heat 0.219 0.051 0.0304 0.094 0.095 0.032 0.054 0.114 0.130 0.031 0.246 0.122 0.109 0.189 0.033 0.183 0.057 0.117 0.203 0.056 0.034 0.096 174 ENGINEERING MATHEMATICS SPECIFIC HEATS (Continued) Liquids Alcohol, methyl Bismuth (melted) Brine (density 1.2) 32° F. Lead (melted) Mercury 68° F Oil, olive Sea- water Sulphur (melted) Tin (melted) Turpentine Water 32° F Water 68° F Water 212° F Specific heat 0.600 0.0363 0.710 0.0402 0.0333 0.47 0.94 0.234 0.064 0.47 1.0083 0.9992 1.0051 Gases Air Ammonia Carbon monoxide Carbon dioxide . . Ethylene Hydrogen Nitrogen Oxygen . Specific heat at constant pressure 0.2375 0.508 0.2479 0.217 0.404 3.409 0.2438 0.2175 Specific heat at constant volume 0.1685 0.299 0.1758 0.171 0.332 2.412 0.1727 0.1550 Other materials Charcoal. . . . Glass, crown. Glass, flint. . Granite Ice India rubber. Marble Masonry Paraffin wax. Porcelain. . . . Quartz Specific heat 0.241 0.16 0.12 0.19 0.504 0.40 0.21 0.20 0.69 0.255 0.18 Note. The specific heat of a material is the number of British Thermal Units necessary' to raise the temperature of 1 pound of the material 1 ° F. PHYSICAL AND CHEMICAL CONSTANTS 1 75 Coefficients of Linear Expansion of Solids The length of a soUd at any temperature is It = lo{l + at), lo being the known length at some given temperature, t the variation of temperature in degrees, and a the coefl&cient of linear expansion of the material. This formula holds approximately when the tempera- ture interval is not large. The coefficient of surface expansion equals 2 a; the coefl&cient of cubical ex- pansion equals 3 a. COEFFICIENTS OF LINEAR EXPANSION (a) Metals Forr C. ForTF. Aluminium 0.0000222 0.000017 0.0000113 0.0000176 0.0000189 0.0000177 0.0000079 0.0000160 0.0000184 0.0000142 0.0000181 0.0000100 0.0000117 0.0000283 0.0000125 0.00000863 0.0000194 0.0000250 0.0000114 0.0000209 0.0000190 0.0000253 0.0000123 Aluminium bronze Antimony 0.0000095 0.00000627 Bismuth 0.00000975 Brass 0.0000105 Bronze 0.00000985 Carbon, graphite 0.0000044 Copper 00000887 German silver (120° F.). • Gold 0.0000102 00000786 Gun metal 0000101 Iron (cast) 00000556 Iron (wrought) 00000648 Lead 0000157 Nickel 00000695 Platinum 0.00000479 Silver 0000108 Solder 0.0000139 Steel 00000636 Tin Type metal (275° F.) Zinc 0.0000116 0.0000106 0.0000141 176 ENGINEERING MATHEMATICS COEFFICIENTS OF LINEAR EXPANSION (a) {Continued) Other materials For 1 ° C. Forl°F. Brick 0.00000550 0.0000143 0.0000770 0.00000850 0.00000714 0.00000812 0.00000789 0.0000507 0.0000040 0.0000060 0.0000036 0.00000050 0.0000104 0.0000030 0.0000050 0.0000600 0.0000400 0.0000340 00000305 Concrete 00000795 Ebonite Glass, soft Glass, hard 0.0000428 0.00000470 0.00000397 Glass, flint 0.00000451 Granite . . 00000438 Ice , 0.0000282 Marble 0.0000022 Masonry (average) Porcelain 0.0000033 0.0000020 Silica (0°to212°F0 Slate 0.00000028 0.00000577 Woods, along grain beech, mahogany oak, pine 0.0000017 0.0000028 Woods, across grain beech 0.0000330 mahogany 0.0000220 pine 0.0000190 ^ PHYSICAL AND CHEMICAL CONSTANTS 77 <u o § « o s <^ m oooofo ro f<^ rs — — CX3 0O0O0O0O — r^vOmfN of vaporiza- tion, L Y tPi '"f (^ OO -^ TfTT oo-^- (N<N<NOvO ovoor^sou-^ ininirMPiin r^^f<^mvOoo rr po «N — o iPiiominm o m sO — o^ o O^O^C30sO^^^ O^ O ' — rsi f<^ — CNfSfNCN — — vOOOt^M OsO — vO — vOOtt oorsi ooooo OOOOO OOOOO OOOOO ;3 1 h 1—1 t>^ fO <N c<^ OO — 0'<r cAOOrr — O o — cN-^r^ c^• o^ o^ o^ o^ oo-^ot^-^ Q\ O^ O^ o^ o^ OOOvO-^eN O^ O^ O^ O^ O^ O^ O pO i/> c^ CMnrrsO — t^mir vOt>* OPor^cNoo ■«T r<^ fN rs — OOOOO r<^ O r>. -^ (N OOOOO lilii pq 1 a 'o O — -^O^vO NOtNu-ivO-^ — t^ — moo ^ go CS <N fS fN rr, fTN c^, cp> (^ cr\ §^^55^5 ^0 — — OmvO sOOOCNO-^ •<9-fNt>.00 OM^^C><^ r^ O^ c^ rO ^■" tj-vCOOOnO §;i^jq^ c«^. fO ^ -"T ^ 1 1 a os-^intsi r^'.r — moo -"f OmvO^ -^OfNO^OO — ■^ — ONO <N|00 — vOC<> vOrOfN — — §2S^S^ OOt^oO-^ — sovou^mm ^^^??i 8^§??2 §§8^2^ i;^^^^^: ^SSi?S:?R ^5t2JQ5^ inoN(NvOOs vONOrN,r>%r^ <NTfr^ON«N OOOOOOOOOn 1 < pressure m inches of mercury .— <Ncc»Trir» vor^oooNO — <Nfc>Trin ^t^^OOO^O 178 ENGINEERING MATHEMATICS s -t-> a hN r^ t>. t>i r>N r^r>.i>«r^ r>, t>i t>i r>. i>. t>. r>i r>. !>• r^ i>. of vaporiza- tion, L r :2- O «NfN(N<N(N fOr<^cric<^ cr> c<^ c«^ CO fO m CO cO m c^ CO do odd <6<6ci<6 d dddod do" odd 1 a t— 1 '— vO — vO<N ooir><NO 00 •— oom-^co COvO — oot^ — c>odsdiX coVi^'d QO oomco — ov hs'roosdco' — oooo oooo O^ C^C^O^O^OO 0O0O0OI>>t^ (y^a^o^o^o^ o^a^ONOv oo oooooooooo oooooooooo ti (N — O^0Or>. vO-^COCN — •— 0^t^>r^CO — OOtOCSOs 1 o o i 'o 'B 'o vooooo— ooovO"^ o ocommco o(NO'«rvO Volume of one pound in cu. ft., U SS2§§ SSS^J? O OCO^^<N O-00<N^ — — — CNCS CNtNfNtN fS «N<NfN(N<N CSJ «N <N CM CS 2- < pressure m inches of mercury p. w PHYSICAL AND CHEMICAL CONSTANTS 179 ctS O o 03 > 1^ ►^lE-. ■3 % a S ■—I s* ^ 03 Q u 05 ^ — sO — t^ro oo-^ — r^-'T oo^o^oooo h>i t>. r* vo ^o t>. vO vO ->0 nO vO vO nO vO sO :S ■^roiporru^ r>, OS — ''I- 1>, oc<^t>. — tn o^cooOf^oo cnfS — 00 00 t>>. r>. sO m tri -^ c<^ r<^ CN — — OOO^ COC<^<^C<^CN rs)(Nrv|<NrS <NfN(N<NfS CNCSCNCN — OS — — O^-^ t^OOoONOr^ 00<N-<rvOt^ tN.sO"^<NON r> c<^ 00 <N t>, — iPiavcor>. o-^t>«.opn vOo^rsimr>i vOr>.t>.000O O^O^OOO — — — rSfS (N<Nc<^c<^f<> co<^r<^cOcO cocOcO'^'^r -"I- -"I- "^ -^ "f 'tr-^'^'^'^ 00000" 00 000 ooo'od cioooo m r>i On rs i>N rs oomt^p^— — — <N-«rtn ooomr^ — Ot>>itn(NO t^wn«^ — OS r>.sOvOsONO unmi/MnTt- 0000000000 0000000000 r>,unc<^'— OS r^tso-^tN — 0000000000 0000000000 — vOCNOstN, imrMnvOt^. OS — "^r^o 0Of<M^<N h^-^fNOsr"* mpo — osr>. m-^cNOOs t>i u^ -"i- rsi — ■^■^'>rfAcr» fOfomtNrq rs^r^p^^r>^— ^_^^_ Os OS O^ Os Os Os Os Os Os Os Os Os Os Os ^^ Os Os Os Os Os r>.os — csm c^rsrsiooo sO "f "-= 00 u^ •— i>.r<^osm sO'>ras'<rvo oooot>.in<N ooc^^r^'— m t>.os'— <s<m oofsmosfN LTiod-— -^r^. — <N<N(NfO c<^ r«^ -«r -^ '"T _ . _ _ fSrsJCSCNCsJ fSrslCNCNCS <NtN<NCNrs) rsl<N<NfS<N lag' som-^om — ■<)-— .r>iso f<^«NorM>.rsi oor>isot^ON r>.os{Nsoo movOfsoo mfNosso-^ — osf^mc^ rKpsjrsj — — ooo^osoo 00 00 i>i r>i r>.* t^isOsOsbsd O) OS rriO\OOs<N CNPslOOO'* O m Os <N m r^ Os Os O O o-^t^on- r>io<^«Xoo" — c<<imooo cs-^vdos — inmmvOso vo t>. t>. r>. r«i ooooooooos Osososo^o <Nrq<NCNCN CN rs| rsj rsj CN ps) rsj psi rsi csi cspsicsipsco o.ti '^ o d-^ . S5 ^ ^ o" < w! o OfN-<fsOQO 0rsl-«r\000 OC^-^sOOO OtN-^sOOO <r\ CO m r<^ cfS '^■•^■^Tr^ UMOmvftvrk ^>OsOnOsO i8o ENGINEERING MATHEMATICS 2 C^ — 0^ sO'^ sOsOsOsOvO sOsOnOvOvO 6131 6113 6096 6079 6062 6045 6028 6012 5996 5981 of vaporiza- tion, L r Oi^r<^ot^ ^ • " ^~ Is o '■*• -^ Tf -<}- TT vOinco — o ooooo OOOOO ooooo ooooo m rt ex OJ o H^ On r>. sO 'Si- CO — oo~^r>.vO pslfS<NrSCS <N(N — OOOOOOOOOO 0O0OO0< or^.'r — oo tPicorq — On \0 "^ ^ O^ OO oot>i\0"^co ooc«^OMn<N oomrqo^i>i -^cnooovo mcAtN — o^ rsi — ooot>* OOOOON 0\ On On 00 OO vomTrrNj— ooNOOt^m O^ O^ O^ O^ O^ O^ OO OO OO OO OOOOOOOOOO OOOOOOOOOO w omomo Tj- a^ c<^ r«N »— inoN(r\r^o Tj-i>.o'^r^ <N(N — OOO vO'^'— oom — i^if^Noo-^ ON-^o^cor*! tN-»f nOOOOn >— f^iTivOOO i>> r*. h*. r> r*. oooooooooo fS<NrN|<Nrvi <NrNj<NCN<N O — cO-^vO <N rj CO CO rr> 2§.s mtnoNr^iOO CNiONONtNoo ONOOr>.\om ■CO<N<N — NONom«AiA mmirMAtn fl) Co o^oot>.irico ot>."^ — 1>. cooOfOooco oor^r^O"^ r^ioNor^co CnKNcOCOCO corocococo cococococo corococOfO ^ 0) rj O dTJ w w d ;4, n w s CT !3^ ^ 2 w P. P, O<N-<rv000 OCnJ'^-nOoO OtNTTsOop SC3^^2S i>» t>» r>» r>N t>» OOOOOOOOOO ^onononon ooooo ^ PHYSICAL AND CHEMICAL CONSTANTS l8l o^ inou^ — ho li-MTMrMrMn cooMntNoo intNot^m (Nooor^in vomrofN — r>i !>«. t^ t^ i>i of vaporiza- tion, L r oo sO ■<*• (^ rsj 2? iO s::? 2^ )fi sss§§ ooooo o on-Or^oO ooooo ooooo OOOOO OOOOO 'S Q. vO-^t^r^JO 0^00 00^>^0 voinmm'.j- TT-^mmin iiiSg O^ O^ O^ O^ Q^ t>. t»» r>. t>« r>. t>i hs r>. t>i t>i ocoooooooo 7 ^< oot>.r^\ONO inmtn-^-^ vOvOt^OOOO rt J-i OOoOOoSoO OOOOOOOOOO g|g|g rrrPitN — O m OJ^sOOO — TfvOO — -"T \oo^ — cAvn r>.o^ — fOtn — u^CXNvO C?^ tN -ra- t>. ON <Ntj-vOOOO «Nf<MnvOI>. fOmc<^roro r<^ co r<^ r<^ r<^ ■<!MnvOt>iOO ^1 B :3 r«*.oo — h^m mvoovom >— <ncoivoo vor^oo^o* inoofNu^cs c<^t>«.(NvO— sO — vo — r>. rsjoO'«roMn oo^o^oot>. r^vONOmm ^j- tj- co co <n rg — — oo ■^mr«^r<^r<^ c^mmcAm c<^f<^f<^r<^m r<\ cr<i trs tr<, cr^ OO — -^hoO ■\t^o<N ■>!»• m r>« oo o ^-cNtTNfA-^ ■^sOh*.QOo — {Nfomsd h>ooo^o(N ro"«rmsor^ po CO c<^ r<^ ■"^ ■^■^■^-^rf TT^-^unm vnu-Momm cocococOfO r<^ CO f<^ ro c<^ cor<^comco c^ co c^ c<> f<^ o stj . P. P* — — — — — rqrsieN(Nrg l82 ENGINEERING MATHEMATICS p. o «o rt o O c t>, vO nO >0 ^ nO sO vO vO 'O tn l/M/MrMJ^i lA ITMO ITMH loinmtnm mirnrM/Mn i/Mommm mmirMrMr> r>i"^fNOtN, intMOoONO f<^ — o^r>»«r> rri — oovO-n- inmmu-*. -^ rf-^j-'ifcoro cot^tNtNCsj cscs — — — ooooo ooooo ooooo ooooo — — — — — — r^^r^^r^^r^^ lA min m u^ irMTM/Mn lA ooo — f<Mn voooo^o — int^iooo^o — rspTMAsO <^r^JtN<Nm fAC<>fAf<>fA iTMniAiAio mmmmiA ooooo ooooo ooooo ooooo pq 2 Q. 1^ vOvOt^t>iOO ONO — (NfA -^J-iAvOOOO^ OtN-^mt^ — c><^ooi>^ vOvOin-^fA fs^o 0^*00 odr^.sOin-^ oo oo t>. t>. h". !>«. r>. r>i t^ h*. t>i t>. r>. \0 vo nOvOsOnOvO t>. i>« r>. r« r>i i>. t>. r>. t>« r>» i>» t>. t>» i>. r>. t>, t>i r>. r^ r>i o^o — (^-^ mvor^o^o (nc^mavooo vO ^^5 vQ \0 ^-O OOOOOOOOOO vOmtriiA«A in, lAtOiAiA OOOOOOOOOO OOOOOOOOOO OCN-'MOt^ ro — oo^od OOOOi &>. t>iO^»— pom t>.OOOfSPA lAvOOOO^^- <N-«*-iAnOOO OOO^O — — (Nr^fAfTNfrs rri f^i (Ts rr> r<^ «NP^1 — — O ,00O>> O— (NfArf' lAvdl^OOOv icAfA ■^'f-^'^'^ -^ •«!»■ "^ -^ ^r I (^ c<\ CO fA cA CO fO fO CO c^ f<^ rn O O^ O^ O^ OO CO <N fsi <N CN OO0Ot>*l^t^ «N <N <N fS <N vO v6 ^ tA«/S CS <N <N r^ CN mm' <N Psi <N <N <N <U TO mmiAvONO vOvOsomm u^-^-^cocm •OO^OOI^ ooa^o — r^ lA lA ■O vO nO CA CA CA CA CA CA CA CA CA CA OOO^O — <N PO-^-^iAvO sOvor>.r^i>« r>. tv. r>. r>i t>i CACACACACA CACACACACA d ^ o :^'d . ti <^ X ^ QtN'n'vOOO QfN-n-sOOO OtN-^vOOO Q <N 3- vO OO lA lA lA lA tA ^o nO vO vO vO t>. t>> l>>. I>« t^ OOOOOOOOOO PHYSICAL AND CHEMICAL CONSTANTS 183 o u <o 0^ ilisl u-mt, vn ir> m r<^ <^ <^ fN (N of vaporiza- tion, L r 00000 <NOt^vOin OvO — t^C<^ —0000 00000 00000 ft f^So^o — •^ rj- ■r^ iTMn ir\ yr\ yr\ lA i/> mm tn vc sO <rNmsOt^^<. sOOOOfN-^ m m m m i/> 00000 00000 00000 00000 1 ^ a; OsO<N'<l-vO OOOOO^O — fNrTMni^ON (Nmt^o-i- W sOvOsOvOsO m mmmm ^^^5^ ^^P^^;?; Ov — <^vO00 O — <Nc<Mn NOOOOfOiTi 00 — "Tr^.- 00 00 00 oo 00 -^fNOOOsO "«r ■<*■ -"a- c<^ c<^ 0000000000 0000000000 0000000000 o^o — p^^"^ inr^o<svr» t<sO^'— f*Mr» noooo — <n t>» 00 00 00 OO 00 00 CT^ O^ O^ O^ O^ O O O O O ^ *" ^^ CT^o^o^o^o^ CTvO^'O^o^^ CT^cooo 00000 — — — — — _____ __(>sjrN)(N «N(N<N<^fS oovoor^vo iAt>.oooo — — — 00 o^r^vo-^c^ OO'— <NfO •^sood — r^' mr>.'o^--fO "^^oos^ iTMO m m lO m «r> in vo >o v© vO vO r>. i*». t^t>»r^oOQO CO c<^ c<^ c<^ r<^ cAfOr<^fOc<^ ror<^cororo rororoc<^f<^ I §.2 m<N — — po "^OsOCSOO OOO^O^OC vo — r^m po <N«NPsJCN<N CNtNtNCsJCN vomcAfNO o^ovooo o^oovomf<>i — OMorsio^ r>.ooo^O— — f<^vdooo — r<MrM>.o — <N-^vOf>. t>» t>« (>• 00 oo 00 00 oO oO O"^ o^ o^ o^ o^ o^ o o o o o c<^ rO CO CO CO CO CO cO CO cO co co co co co ^ ■^ ^ ■^ rf <u.S S3 o md . p. p« OfS-'J-NOOO Otr»OiA< O^ C^ O^ CT^ On O O "— ^ < Ovoomo CNcOCO-^-er «rMONO>Ot>* i84 ENGINEERING MATHEMATICS 2 K|E-< OS o r>.mcofNo o^sOro —.— — — — ooo in i/^ lA ir> un iTMn u^ O^ ^^ O^ O^ O^ O^ ^^ O*^ O^ OO oooooooooo oooooor>it>. ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo S o (N — o^ r>i sO '«»■— od in rsi o^ sO -^ — od m m o od t>. r<^m<N<N«N rvirsl — — — OOOOO O^OOt>."^fS r^ r^ 1**^ r^ t^ r^ r^ t^ r*% r*% 1*% t^ r^ t^ ^<o ^^o ^^3 ^^o ^'O ^^5 ■"ToocNvoo ■^■^(^m-^ msooo — en so-^ooom r^inrr^N— o^vdcnor*. ■^'— odvom or^m — o — .— — — — ooooo o^o^oooooo oc vO m cn — oooocx)ooco oooooooor*. t^ r^, r>. t>. t>. h*. t>, t>. i>. r>. t3 ■«f in vO tN, CO ovofscn-^ inm^ovOvO mrm^oo'^ — — — — — — fses<NtN ooooo ooooo fNfSCNfNCS tN fS fS fN r^ OOOOO oooo^o^ <:>ts,Tj- — OO -'i-h^ia^oo mm r>iO>> o 00 0O0OC30ON rn cn cn cn cn c^i in OO fs in ooM>.infn OO o cn vO o^ OOOvOO — <NTj-vOOOt>. •Si O 3 a ^ a; ^ in-^-'T -^cn cn<Nr>4«N— — OO^t>.vO sOr^ioo-^o mm ^cn ONvO<Noocn oomtNm-^ ■«t- vO t>.' vd cn o^ — (N'^in r>.ocnvooN — ■'j-rs.ovcN ■«»■ vO t>. vO cn o — — — — ■ — cN(N(Nc^i cncncncn-^ -^m-oooo ■^■^'tr-n-'^ ■^■"T'^'^'^ ■^■^'^■^•■^ TT-^TTTrm w d momom oooOQ ooooo QOQCQ r<»ooooo^a^ o — f^cn^ m^or^iooo^ QmoQO (NfNrs<NCN c^mcncncn cncncncncn ■^■«rmvor>» ^^ L '850 900 950 100O HEAT CONTENT, B.T.U. 1050 MOLLIER S Reproduced with permission from G. A. Goodenoug' Messrs. Joh( 13S0 1450. 2.10 1200 1250 1300 1390 1400 1450 1.30 AM CHART "Properties of Steam and Ammonia," published by iley & Sons. i w TABLES I8S TABLES CIRCUMFERENCES AND AREAS OF CIRCLES ' Diam- Circum- Area Diam- Circum- Area eter ference eter ference 1 3.1416 0.7854 26 81.681 530.93 2 6 2832 3.1416 27 84.823 572.56 3 9.4248 7.0686 28 87.965 615.75 4 12.5664 12.5664 29 91.106 660.52 5 15.7080 19.635 30 94.248 706.86 6 18.850 28.274 31 97.389 754.77 7 21.991 38.485 32 100.53 804.25 8 25.133 50.266 33 103.67 855.30 9 28.274 63.617 34 106.81 907.92 10 31.416 78.540 35 109.96 962.11 11 34.558 95.033 36 113.10 1017.88 12 37.699 113.10 37 116.24 1075.21 13 40.841 132.73 38 119.38 1134.11 14 43.982 153.94 39 122.52 1194.59 15 47.124 176.71 40 125.66 1256.64 16 50.265 201.06 41 128.81 1320.25 17 53.407 226.98 42 131.95 1385.44 18 56.549 254.47 43 135.09 1452.20 19 59.690 283.53 44 138.23 1520.53 20 62.832 314.16 45 141.37 1590.43 21 65.973 346.36 46 144.51 1661 .90 22 69.115 380.13 47 147.65 1734.94 23 72.257 415.48 48 150.80 1809.56 24 75.398 452.39 49 153.94 1885.74 25 78.540 490.87 50 157.08 1963.50 Note. — The surface of a sphere of given diameter may be found directly from the above table, since it is equal to the area of a circle of twice the diam- eter of the sphere. i86 ENGINEERING MATHEMATICS CIRCUMFERENCES AND AREAS OF CIRCLES ( Continued) Diam- eter 51 52 53 54 55 56 5.7 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 Circum- ference 160.22 163.36 166.50 169.65 172.79 175.93 179.07 182.21 185.35 188.50 191.64 194.78 197.92 201.06 204.20 207.34 210.49 213.63 216.77 219.91 223.05 226.19 229.34 232.48 235.62 Area 2042.82 2123.72 2206.18 2290.22 2375.83 2463.01 2551.76 2642.08 2733.97 2827.43 2922.47 '3019.07 3117.25 3216.99 3318.31 3421.19 3525.65 3631.68 3739.28 3848.45 3959 . 19 4071.50 4185.39 4300.84 4417.86 Diam- Circum- eter ference 76 238.76 77 241.90 78 245.04 79 248 . 19 80 251.33 81 254.47 82 257.61 83 260.75 84 263.89 85 267 .04 86 270.18 87 273.32 88 276.46 89 279.60 90 282.74 91 285.88 92 289.03 93 292.17 94 295.31 95 298.45 96 301.59 97 304.73 98 307.88 99 311.02 100 314.16 Area 4536.46 4656.63 4778.36 4901.67 5026.55 5153.00 5281 .02 5410.61 5541.77 5674.50 5808.80 5944.68 6082.12 6221.14 6361.73 6503.88 6647.61 6792.91 6939.78 7088.22 7238.23 7389.81 7542.96 7697.69 7853.98 TABLES 187 POWERS, ROOTS, AND RECIPROCALS Number Square Cube Square root Cube root Reciprocal 1 1 1 1.000000 1.000000 1.0000000 2 4 8 1.414214 1.259921 .5000000 3 9 27 1.732051 1.442250 .3333333 4 16 64 2.000000 1.587401 .2500000 5 25 125 2.236068 1.709976 .2000000 6 36 216 2.449490 1.817121 .1666667 7 49 343 2.645751 1.912931 .1428571 8 64 512 2.828427 2.000000 . 1250000 9 81 729 3.000000 2.080084 .1111111 10 100 1000 3.162278 2 . 154435 .1000000 11 121 1331 3.316625 2.223980 .0909091 12 144 1728 3.464102 2.289429 m33333 13 169 2197 3.605551 2.351335 .0769231 14 196 2744 3.741657 2.410142 .0714286 15 225 3375 3.872983 2.466212 .0666667 16 256 4096 4.000000 2.519842 .0625000 17 289 4913 4.123106 2.571282 .0588235 18 324 5832 4.242641 2.620741 .0555556 19 361 6859 4.358899 2.668402 .0526316 20 400 8000 4.472136 2.714418 .0500000 21 441 9261 4.582576 2.758924 .0476190 22 484 10,648 4.690416 2.802039 .0454545 23 529 12,167 4.795832 2.843867 .0434783 24 576 13,824 4.898980 2.884499 .0416667 25 625 15,625 5.000000 2.924018 .0400000 26 676 17,576 5.099020 2.962496 .0384615, 27 729 19,683 5.196152 3.000000 .0370370 28 784 21,952 5.291503 3.036589 .0357143 29 841 24,389 5.385165 3.072317 .0344828 30 900 27,000 5.477226 3.107233 .0333333 31 961 29,791 5.567764 3.141381 .0322581 32 1024 32,768 5.656854 3.174802 .0312500 33 1089 35,937 5.744563 3.207534 .0303030 34 1156 39,304 5.830952 3.239612 .0294118 35 1225 42,875 5.916080 3.271066 .0285714 36 1296 46,656 6.000000 3.301927 .0277778 37 1369 50,653 6.082763 3.332222 .0270270 i88 ENGINEERING MATHEMATICS POWERS, ROOTS, AND RECIPROCALS {Continued) Number Square Cube Square root Cube root Reciprocal 38 1444 54,872 6.164414 3.361975 .0263158 39 1521 59,319 6.244998 3.391211 .0256410 40 1600 64,000 6.324555 3.419952 .0250000 41 1681 68,921 6.403124 3.448217 .0243902 42 1764 74,088 6.480741 3.476027 .0238095 43 1849 79,507 6.557439 3.503398 .0232558 44 1936 85,184 6.633250 3.530348 .0227273 45 2025 91,125 6.708204 3.556893 .0222222 46 2116 97,336 6.782330 3.583048 .0217391 47 2209 103,823 6.855655 3.608826 .0212766 48 2304 110,592 6.928203 3.634241 .0208333 49 2401 117,649 7.000000 3.659306 .0204082 50 2500 125,000 7.071068 3.684031 .0200000 51 2601 132,651 7.141428 3.708430 .0196078 52 2704 140,608 7.211103 3.732511 .0192308 53 2809 148,877 7.280110 3.756286 .0188679 54 2916 157,464 7.348469 3.779763 .0185185 55 3025 166,375 7.416199 3.802953 .0181818 56 3136 175,616 7.483315 3.825862 .0178571 57 3249 185,193 7.549834 3.848501 .0175439 58 3364 195,112 7.615773 3.870877 .0172414 59 3481 205,379 7.681146 3.892997 .0169492 60 3600 216,000 7.745967 3.914868 .0166667 61 3721 226,981 7.810250 3.936497 .0163934 . 62 3844 238,328 7.874008 3.957892 .0161290 63 3969 250,047 7.937254 3.979057 .0158730 64 4096 262,144 8.000000 4.000000 .0156250 65 4225 274,625 8.062258 4.020726 .0153846 66 4356 287,496 8.124038 4.041240 .0151515 67 4489 300,763 8.185353 4.061548 .0149254 68 4624 314,432 8.246211 4.081655 .0147059 69 4761 328,509 8.306624 4.101566 .0144928 70 4900 343,000 8.366600 4.121285 .0142857 71 5041 357,911 8.426150 4.140818 .0140845 72 5184 373,248 8.485281 4.160168 .0138889 73 5329 389,017 8.544004 4.179339 .0136986 TABLES 189 POWERS, ROOTS, AND RECIPROCALS (Continued) Number Square Cube Square root Cube root Reciprocal 74 5476 405.224 8.602325 4 . 198336 .0135135 75 5625 421,875 8.660254 4.217163 .0133333 76 5776 438,976 8.717798 4.235824 .0131579 77 5929 456,533 8.774964 4.254321 .0129870 78 6084 474,552 8.831761 4.272659 .0128205 79 6241 493,039 8.888194 4.290840 .0126582 80 6400 512,000 8.944272 4.308870 .0125000 81 6561 531,441 9.000000 4.326749 .0123457 82 6724 551,368 9.055385 4.344482 .0121951 83 6889 571,787 9.110434 4.362071 .0120482 84 7056 592,704 9.165151 4.379519 .0119048 85 7225 614,125 9.219545 4.396830 .0117647 86 7396 636,056 9.273619 4.414005 .0116279 87 7569 658,503 9.327379 4.431048 .0114943 88 7744 681,472 9.380832 4.447960 .0113636 89 7921 704,969 9.433981 4.464745 .0112360 90 8100 729,000 9.486833 4.481405 .0111111 91 8281 753,571 9.539392 4.497941 .0109890 92 8464 778,688 9.591663 4.514357 0108696 93 8649 804,357 9.643651 4.530655 .0107527 94 8836 830,584 9.695360 4.546836 .0106383 95 9025 857,375 9.746794 4.562903 .0105263 96 . 9216 884,736 9.797959 4.578857 .0104167 97 9409 . 912,673 9.848858 4.594701 .0103093 98 9604 941,192 9.899495 4.610436 .0102041 99 9801 970,299 9.949874 4.626065 .0101010 100 10,000 1,000,000 10.000000 4.641589 .0100000 Logarithmic Cross- section Paper Cross-section paper the rulings of which are pro- portional to the logarithms of the scale is called loga- rithmic cross-section paper. This paper is most con- venient for plotting equations with constant exponents since they are straight lines on logarithmic paper while 190 ENGINEERING MATHEMATICS 'they are curves if plotted on ordinary graph paper, in which case they must be plotted point by point. The chief use of logarithmic cross-section paper is for plotting equations of the form: y = ax^ If two pairs of values of x and y are known, the corre- sponding points may be plotted on logarithmic paper and joined by a straight line. The value of the coeffi- cient a is equal to the intercept of this Une on the F-axis, and the value of the exponent n is equal to the slope of the line (that is, the tangent of the angle which the line makes with the X-axis). The reason for this is that plotting on logarithmic paper is equivalent to taking logarithms, in which case we would obtain: log y = loga + n log x which is the equation of a straight line, log a being the intercept and n the slope. In case the values of a and n are known, that is, the intercept and the slope, we may plot the line, and from it obtain any pair of values of x and y. Use of Logarithm Tables Every logarithm consists of two parts: a positive or negative whole number called the characteristic, and a positive fraction, called the mantissa. The mantissa is always expressed as a decimal, and is the part which is given in the tables. To find the common logarithm of a given number: If the number is greater than 1, the characteristic of the logarithm is one unit less than the number of fig- ures on the left of the decimal point. TABLES 191 If the number is less than 1, the characteristic of the logarithm is negative, and one unit more than the number of zeros between the decimal point and the first significant figure of the given nimiber. Thus, log 20.6 = 1.3139 (base 10) log 2.06 = 0.3139 log 0.206 = 0.3139 - 1 = 9.3139 - 10 log 0.0206 = 0.3139 - 2 = 8.3139 - 10 To find the number corresponding to a given common logarithm : If the characteristic of a given logarithm is posi- tive, the number of figures in the integral part of the corresponding number is one more than the number of units in the characteristic. If the characteristic is negative, the number of zeros between the decimal point and the first significant figure of the corresponding number is one less than the number of units in the characteristic. 192 ENGINEERING MATHEMATICS COMMON LOGARITHMS OF NUMBERS (Base 10) N 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 . 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6??? 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 TABLES 193 COMMON LOGARITHMS OF NUMBERS (Continued) N 1 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9599 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 194 ENGINEERING MATHEMATICS NATURAL LOGARITHMS OF NUMBERS FROM 1 TO 10 (Base e) 0.0000 0.0953 0.1823 0.2624 0.3365 0.4055 0.4700 0.5306 0.5878 0.6419 0.0099 0.1044 0.1906 0.2700 0.3436 0.4121 0.4762 0.5365 0.5933 0.6471 0.6932 0.6981 0.7419 0.7467 0.7885 0.8329 0.8755 0.9163 0.9555 0.9933 1.0296 1.0647 1.0986 1.1314 1.1632 1.1939 1.2238 1.2528 1.2809 1.3083 1.3350 1.3610 1.3863 1.4110 1.4351 1.4586 1.4816 1.5041 1.5261 1.5476 1.5686 1.5892 1.6094 1.6292 1.6487 1.6677 1.6864 0.7950 0.8373 0.8796 0.9203 0.9594 0.9970 1.0332 1.0681 1.1019 1.1346 1.1663 1.1970 1.2267 1.2556 1.2837 1.3110 1.3376 1.3635 1.3888 1.4134 1 4375 1.4609 1.4839 1.5063 1.5282 1.5497 1.5707 1.5913 1.6114 1.6312 1.6506 0.0198 0.1133 0.1989 0.2776 0.3507 0.4187 0.4824 0.5423 0.5988 0.6523 0.7031 0.7514 0.7975 0.8416 0.8838 0.9243 0.9632 1.0006 1.0367 1.0716 1.1053 1.1378 1 1694 1.2000 1.2296 1.2585 1.2865 1.3137 1.3403 1.3661 1.3913 1.4159 1.4398 1.4633 1.4861 1.5085 1.5304 1.5518 1.5728 1.5933 1.6134 1.6332 1.6525 1.6696 1.6715 1.6883 1.6901 0.0296 0.1222 0.2070 0.2852 0.3577 0.4253 0.4886 0.5481 0.6043 0.6575 0.7080 0.7561 0.8020 0.8459 0.8879 0.9282 0.9670 1.0043 1 . 0403 1.0750 1.1086 1.1410 1.1725 1.2030 1.2326 1.2613 1 . 2892 1.3164 1.3429 1.3686 1.3938 1.4183 1.4422 1.4656 1.4884 1.5107 1.5326 1.5539 1.5749 1.5953 1.6154 1.6351 1.6545 1.6734 1.6919 0.0392 0.1310 0.2151 0.2927 0.3646 0.4318 0.4947 0.5539 0.6098 0.6627 0.7130 0.7608 0.8065 0.8502 0.8920 0.9322 0.9708 1.0080 1.0438 1.0784 1.1119 1.1442 1.1756 1.2060 1.2355 1.2641 1.2920 1.3191 1.3455 1.3712 1.3962 1.4207 1.4446 1.4679 1.4907 5129 5347 55^0 5769 1 1 1 1 1.5974 1.6174 1.6371 1.6563 1.6753 1.6938 0.0488 0.1398 0.2231 0.3001 0.3716 0.4383 0.5008 0.5596 0.6152 0.6678 0.7178 0.7655 0.8109 0.8544 0.8961 0.9361 0.9746 1.0116 1.0473 1.0818 1.1151 1.1474 1.1787 1.2090 1.2384 1.2670 1.2947 1.3218 1.3481 1.3737 1.3762 1.3987 1.4231 1.4469 1.4702 1.4929 1.5151 1.5369 1.5581 1.5790 1.5994 1.6194 1.6390 1.6582 1.6771 1.6956 0.0583 0.1484 0.2311 0.3075 0.3784 0.4447 0.5068 0.5653 0.6206 0.6729 0.7227 0.7701 0.8154 0.8587 0.9001 0.9400 0.9783 1.0152 1.0508 1.0852 1.1184 1.1506 1.1817 1.2119 1.2413 1.2698 1.2975 1.3244 1.3507 1.4012 1.4255 1.4493 1.4725 1.4951 1.5173 1.5390 1.5603 1.5810 1.6014 1.6214 1.6409 1.6601 1.6790 1.6975 0.0677 0.1570 0.2390 0.3148 0.3853 0.4511 0.5128 0.5710 0.6258 0.6780 0.7276 0.7747 0.8198 0.8629 0.9042 0.9439 0.9820 1.0189 1.0543 1.0886 1.1217 1.1537 1.1848 1.2149 1.2442 1.2726 1.3002 1.3271 1.3533 1.3788 1.4036 1.4279 1.4516 1.4748 1.4974 1.5195 1.5412 1.5624 1.5831 1.6034 1.6233 1.6429 1.6620 1.6808 1.6993 0.0770 0.1655 0.2469 0.3221 0.3920 0.4574 0.5188 0.5766 0.6313 0.6831 0.7324 0.7793 0.8242 0.8671 0.9083 0.9478 0.9858 1.0225 1.0578 1.0919 1.1249 1.1569 1.1878 1-2179 1.2470 1.2754 1.3029 1.3297 1.3558 1.3813 1.4061 1.4303 1.4540 1.4770 1.4996 1.5217 1.5433 1.5644 1.5852 1.6054 1.6253 1.6448 1.6639 1.6827 1.7011 0.0862 0.1740 0.2546 0.3293 0.3988 0.4637 0.5247 0.5822 0.6366 0.6881 0.7372 0.7839 0.8286 0.8713 0.9123 0.9517 0.9895 1.0260 1.0613 1.0953 1.1282 1 . 1600 1.1909 1.2208 1.2499 1.2782 1.3056 1.3324 1.3584 1.3838 1.4085 1.4327 1.4563 1.4793 1.5019 1.5239 1.5454 1.5665 1.5872 1.6074 1.6273 1.6467 1.6658 1.6846 1.7029 TABLES 195 NATURAL LOGARITHMS OF NUMBERS {Continued) N 1 2 3 4 5 6 7 8 9 5.5 1.7048 1.7066 1.7084 1.7102 1.7120 1.7138 1.7156 1.7174 1.7192 1.7210 5.6 1.7228 1.7246 1.7263 1.7281 1.7299 1.7317 1.7334 1.7352 1.7370 1.7387 5.7 1.7405 1.7422 1.7440 1.7457 1.7475 1.7491 1.7509 1.7527 1.7544 1.7561 5.8 1.7579 1.7596 1.7613 1.7630 1.7647 1.7664 1.7682 1.7699 1.7716 1.7733 5.9 1.7750 1.7767 1.7783 1.7800 1.7817 1.7834 1.7851 1.7868 1.7884 1.7901 6.0 1.7918 1.7934 1.7951 1.7968 1.7984 1.8001 1.8017 1.8034 1.8050 1.8067 6.1 1.8033 1.8099 1.8116 1.8132 1.8148 1.8165 1.8181 1.8197 1.8213 1.8229 6.2 1.8246 1.8262 1.8278 1.8294 1.8310 1.8326 1.8342 1.8358 1.8374 1.8390 6.3 1.8406 1.8421 1.8437 1.8453 1.8469 1.8485 1.8500 1.8516 1.8532 1.8547 6.4 1.8563 1.8579 1.8594 1.8610 1.8625 1.8641 1.8656 1.8672 1.8687 1.8703 6.5 1.8718 1.8733 1.8749 1.8764 1.8779 1.8795 1.8810 1.8825 1.8840 1.8856 6.6 1.8871 1.8886 1.8901 1.8916 1.8931 1.8946 1.8961 1.8976 1.8991 1.9006 6.7 1.9021 1.9036 1.9051 1.9066 1.9081 1.9095 1.9110 1.9125 1.9140 1.9155 6.8 1.9169 1.9184 1.9199 1.9213 1.9228 1.9243 1.9257 1.9272 1.9286 1.9301 6.9 1.9315 1.9330 1.9344 1.9359 1.9373 1.9387 1.9402 1.9416 1.9431 1.9445 7.0 1.9459 1.9473 1.9488 1.9502 1.9516 1.9530 1.9545 1.9559 1.9573 1.9587 7.1 1.9601 1.9615 1.9629 1.9643 1.9657 1.9671 1.9685 1.9699 1.9713 1.9727 7.2 1.9741 1.9755 1.9769 1.9782 1.9796 1.9810 1.9824 1.9838 1.9851 1.9865 7.3 1.9879 1.9892 1.9906 1.9920 1.9933 1.9947 1.9961 1.9974 1.9988 2.0001 7.4 2.0015 2.0028 2.0042 2.0055 2.0069 2.0082 2.0096 2.0109 2.0122 2.0136 7.5 2.0149 2.0162 2.0176 2.0189 2.0202 2.0216 2.0229 2.0242 2.0255 2.0268 7.6 2.0282 2.0295 2.0308 2.0321 2.0334 2.0347 2.0360 2.0373 2.0386 2.0399 7.7 2.0412 2.0425 2.0438 2.0451 2.0464 2.0477 2.0490 2.0503 2.0516 2.0528 7.8 2.0541 2.0554 2.0567 2.0580 2.0592 2.0605 2.0618 2.0631 2.0643 2.0656 7.9 2.0669 2.0681 2.0694 2.0707 2.0719 2.0732 2.0744 2.0757 2.0769 2.0782 8.0 2.0794 2.0807 2.0819 2.0832 2.0844 2.0857 2.0869 2.0882 2.0894 2.0906 8.1 2.0919 2.0931 2.0943 2.0956 2.0968 2.0980 2.0992 2.1005 2.1017 2.1029 8.2 2.1041 2.1054 2.1066 2.1078 2.1090 2.1102 2.1114 2.1126 2.1138 2.1151 8.3 2.1163 2.1175 2.1187 2.1199 2.1211 2.1223 2.1235 2.1247 2.1259 2.1270 8.4 2.1282 2.1294 2.1306 2.1318 2.1330 2.1342 2.1354 2.1365 2.1377 2.1389 8.5 2.1401 2.1412 2.1424 2.1436 2.1448 2.1459 2.1471 2.1483 2.1494 2.1506 8.6 2.1518 2.1529 2.1541 2.1552 2.1564 2.1576 2.1587 2.1599 2.1610 2.1622 8.7 2.1633 2.1645 2.1656 2.1668 2.1679 2.1691 2.1702 2.1713 2.1725 2.1736 8.8 2.1748 2.1759 2.1770 2.1782 2.1793 2.1804 2.1816 2.1827 2.1838 2.1849 8.9 2.1861 2.1872 2.1883 2.1894 2.1905 2.1917 2.1928 2.1939 2.1950 2.1961 9.0 2.1972 2.1983 2.1994 2.2006 2.2017 2.2028 2.2039 2.2050 2.2061 2.2072 9.1 2.2083 2.2094 2.2105 2.2116 2.2127 2.2138 2.2149 2.2159 2.2170 2.2181 9 2 2.2192 2.2203 2.2214 2.2225 2.2235 2.2246 2.2257 2.2268 2.2279 2.2289 9.3 2.2300 2.2311 2.2322 2.2332 2.2343 2.2354 2.2365 2.2375 2.2386 2.2397 9.4 2.2407 2.2418 2.2428 2.2439 2.2450 2.2460 2.2471 2.2481 2.2492 2.2502 9.5 2.2513 2.2523 2.2534 2.2544 2.2555 2.2565 2.2576 2.2586 2.2597 2.2607 9.6 2.2618 2.2628 2.2638 2.2649 2.2659 2.2570 2.2680 2.2690 2.2701 2.2711 9.7 2.2721 2.2732 2.2742 2.2752 2.2762 2.2773 2.2783 2.2793 2.2803 2.2814 9.8 2.2824 2.2834 2.2844 2.2854 2.2865 2.2875 2.2885 2.2895 2.2905 2.2915 9.9 2.2925 2.2935 2.2946 2.2956 2.2966 2.2976 2.2986 2.2996 2.3006 2.3016 196 ENGINEERING MATHEMATICS NATURAL LOGARITHMS (EACH INCREASED BY 10) OF NUMBERS FROM 0.00 TO 0.99 No. 1 2 3 4 5 6 7 8 9 0.0 5.395 6.088 6.493 6.781 7.004 7.187 7.341 7.474 7.592 0.1 7.697 7.793 7.880 7.960 8.034 8.103 8.167 8.228 8.285 8.339 0.2 8.391 8.439 8.486 8.530 8.573 8.614 8.653 8.691 8.727 8.762 0.3 8.796 8.829 8.861 8.891 8.921 8.950 8.978 9.006 9.032 9.058 0.4 9.084 9.108 9.132 9.156 9.179 9.201 9.223 9.245 9.266 9.287 0.5 9.307 9.327 9.346 9.365 9.384 9.402 9.420 9.438 9.455 9.472 0.6 9.489 9.506 9.522 9.538 9.554 9.569 9.584 9.600 9.614 9.629 0.7 9.643 9.658 9.671 9.685 9.699 9.712 9.726 9.739 9.752 9.764 0.8 9.777 9.789 9.802 9.814 9.826 9.837 9.849 9.861 9.872 9.883 0.9 9.895 9.906 9.917 9.927 9.938 9.949 9.959 9.970 9.980 9.990 NATURAL LOGARITHMS OF WHOLE NUMBERS FROM 10 TO 209 No. 1 2 3 4 5 6 7 8 9 1 2.303 2.398 2.485 2.565 2.639 2.708 2.773 2.833 2.890 2.944 2 2.996 3.045 3.091 3.136 3.178 3.219 3.258 3.296 3.332 3.367 3 3.401 3.434 3.466 3.497 3.526 3.555 3.584 3.611 3.638 3.664 4 3.689 3.714 3.738 3.761 3.784 3.807 3.829 3.850 3.871 3.892 5 3.912 3.932 3.951 3.970 3.989 4.007 4.025 4.043 4.060 4.078 6 4.094 4.111 4.127 4.143 4.159 4.174 4.190 4.205 4.220 4.234 7 4.249 4.263 4.277 4.291 4.304 4.318 4.331 4.344 4.357 4.369 8 4.382 4.394 4.407 4.419 4.431 4.443 4.454 4.466 4.477 4.489 9 4.500 4.511 4.522 4.533 4.543 4.554 4.564 4.575 4.585 4.595 10 4.605 4.615 4.625 4.635 4.644 4.654 4.663 4.673 4.682 4.691 11 4.701 4.710 4.719 4.727 4.736 4.745 4.754 4.762 4.771 4.779 12 4.788 4.796 4.804 4.812 4.820 4.828 4.836 4.844 4.852 4.860 13 4.868 4.875 4.883 4.890 4.898 4.905 4.913 4.920 4.927 4.935 14 4.942 4.949 4.956 4.963 4.970 4.977 4.984 4.990 4.997 5.004 15 5.011 5.017 5.024 5.030 5.037 5.043 5.050 5.056 5.063 5.069 16 5.075 5.081 5.088 5.094 5.100 5.106 5.112 5.118 5.124 5.130 17 5.136 5.142 5.148 5.153 5.159 5.165 5.171 5.176 5.182 5.187 18 5.193 5.199 5.204 5.210 5.215 5.220 5.226 5.231 5.236 5.242 19 5.247 5.252 5.258 5.263 5.268 5.273 5.278 5 283 5.288 5.293 20 5.298 5.303 5.308 5.313 5.318 5.323 5.328 5.333 5.338 5.342 TABLES 197 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS Degrees sin COS tan cot 0°00' — 00 10.0000 — 00 + 00 90° 00' 0° 10' 7.4637 9.9999 7.4637 2.5363 89° 50' 0°20' 7.7648 9.9999 7.7648 2.2352 89° 40' 0° 30' 7.9408 9.9999 7.9409 2.0591 89° 30' 0°40' 8.0658 9.9999 8.0658 1.9342 89° 20' 0°50' 8.1627 9.9999 8.1627 1.8373 89° 10' r 00' 8.2419 9.9999 8.2419. 1.7581 89° 00' 1° 10' 8.3088 9.9999 8.3089 1.6911 88° 50' 1°20' 8.3668 9.9999 8.3669 1.6331 88° 40' 1°30' 8.4179 9.9999 8.4181 1.5819 88° 30' 1°40' 8.4637 9.9998 8.4638 1.5362 88° 20' 1°50' 8.5050 9.9998 8.5053 1.4947 88° 10' 2° 00' 8.5428 9.9997 8.5431 1.4569 88° 00' 2° 10' 8.5776 9.9997 8.5779 1.4221 87° 50' 2° 20' 8.6097 9.9996 8.6101 1.3899 87° 40' 2° 30' 8.6397 9.9996 8.6401 1.3599 87° 30' 2° 40' 8.6677 9.9995 8.6682 1.3318 87° 20' 2° 50' 8.6940 9.9995 8.6945 1.3055 87° 10' 3° 00' 8.7188 9.9994 8.7194 1.2806 87° 00' 3° 10' 8.7423 9.9993 8.7429 1.2571 86° 50' 3° 20' 8.7645 9.9993 8.7652 1.2348 86° 40' 3° 30' 8.7857 9.9992 8.7865 1.2135 86° 30' 3° 40' 8.8059 9.9991 8.8067 1 . 1933 86° 20' 3° 50' 8.8251 9.9990 8.8261 1.1739 86° 10' 4° 00' 8.8436 9.9989 8.8446 1.1554 86° 00' 4° 10' 8.8613 9.9989 8.8624 1.1376 85° 50' 4° 20' 8.8783 9.9988 8.8795 1.1205 85° 40' 4° 30' 8.8946 9.9987 8.8960 1.1040 85° 30' 4° 40' 8.9104 9.9986 8.9118 1.0882 85° 20' 4° 50' 8.9256 9.9985 8.9272 1.0728 85° 10' 5° 00' 8.9403 9.9983 8.9420 1.0580 85° 00' 5° 10' 8.9545 9.9982 8.9563 1.0437 84° 50' 5° 20' 8.9682 9.9981 8.9701 1.0299 84° 40' 5° 30' 8.9816 9.9980 8.9836 1.0164 84° 30' cos sin cot tan Degrees 198 ENGINEERING MATHEMATICS LOGARITHMIC wSINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 5° 40' 8.9945 9.9979 8.9966 1.0034 84° 20' 5° 50' 9.0070 9.9977 9.0093 0.9907 84° 10' 6° 00' 9.0192 9.9976 9.0216 0.9784 84° 00' 6° 10' 9.0311 9.9975 9.0336 0.9664 83° 50' 6° 20' 9.0426 9.9973 9.0453 0.9547 83° 40' 6° 30' 9.0539 9.9972 9.0567 0.9433 83° 30' 6° 40' 9.0648 9.9971 9.0678 0.9322 83° 20' 6° 50' 9.0755 9.9969 9.0786 0.9214 83° 10' 7° 00' 9.0859 9.9968 9.0891 0.9109 83° 00' 7° 10' • 9.0961 9.9966 9.0995 0.9005 82° 50' 7° 20' 9.1060 9.9964 9.1096 0.8904 82° 40' 7° 30' 9.1157 9.9963 9.1194 0.8806 82° 30' 7° 40' 9.1252 9.9961 9.1291 0.8709 82° 20' 7° 50' 9.1345 9.9959 9.1385 0.8615 82° 10' 8° 00' 9.1436 9.9958 9.1478 0.8522 82° 00' 8° 10' 9.1525 9.9956 9.1569 0.8431 81° 50' 8° 20' 9.1612 9.9954 9.1658 0.8342 81° 40' 8° 30' 9 . 1697 9.9952 9.1745 0.8255 81° 30' 8° 40' 9.1781 9.9950 9.1831 0.8169 81° 20' 8° 50' 9.1863 9.9948 9.1915 0.8085 81° 10' 9° 00' 9 . 1943 9.9946 9.1997 0.8003 81° 00' 9° 10' 9.2022 9.9944 9.2078 0.7922 80° 50' 9° 20' 9.2100 9.9942 9.2158 0.7842 80° 40' 9° 30' 9.2176 9.9940 9.2236 0.7764 80° 30' 9° 40' 9.2251 9.9938 9.2313 0.7687 80° 20' 9° 50' 9.2324 9.9936 9.2389 0.7611 80° 10' 10° 00' 9.2397 9.9934 9.2463 0.7537 80° 00' 10° 10' 9.2468 9.9931 9.2536 0.7464 79° 50' 10° 20' 9.2538 9.9929 9.2609 0.7391 79° 40' 10° 30' 9.2606 9.9927 9.2680 0.7320 79° 30' 10° 40' 9.2674 9.9924 9.2750 0.7250 79° 20' 10° 50' 9.2740 9.9922 9.2819 0.7181 79° 10' 11° 00' 9.2806 9.9919 9.2887 0.7113 79° 00' 11° 10' 9.2870 9.9917 9.2953 0.7047 78° 50' cos sin cot tan Degrees TABLES 199 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS (Continued) Degrees sin COS tan cot 11° 20' 9.2934 9.9914 9.3020 0.6980 78° 40' 11° 30' 9.2997 9.9912 9.3085 0.6915 78° 30' 11° 40' 9.3058 9.9909 9.3149 0.6851 78° 20' 11° 50' 9.3119 9.9907 9.3212 0.6788 78° 10' 12° 00' 9.3179 9.9904 9.3275 0.6725 78° 00' 12° 10' 9.3238 9.9901 9.3336 0.6664 77° 50' 12° 20' 9.3296 9.9899 9.3397 0.6603 77° 40' 12° 30' 9.3353 9.9896 9.3458 0.6542 77° 30' 12° 40' 9.3410 9.9893 9.3517 0.6483 77° 20' 12° 50' 9.3466 9.9890 9.3576 0.6424 77° 10' 13° 00' 9.3521' 9.9887 9.3634 0.6366 77° 00' 13° 10' 9.3575 9.9884 9.3691 0.6309 76° 50' 13° 20' 9.3629 9.9881 9.3748 0.6252 76° 40' 13° 30' 9.3682 9.9878 9.3804 0.6196 76° 30' 13° 40' 9.3734 9.9875 9.3859 0.6141 76° 20' 13° 50' 9.3786 9.9872 9.3914 0.6086 76° 10' 14° 00' 9.3837 9.9869 9.3968 0.6032 76° 00' 14° 10' 9.3887 9.9866 9.4021 0.5979 75° 50' 14° 20' 9.3937 9.9863 9.4074 0.5926 75° 40' 14° 30' 9.3986 9.9859 9.4127 0.5873 75° 30' 14° 40' 9.4035 9.9856 9.4178 0.5822 75° 20' 14° 50' 9.4083 9.9853 9.4230 0.5770 75° 10' 15° 00' 9.4130 9.9849 9.4281 0.5719 75° 00' 15° 10' 9.4177 9.9846 9.4331 0.5669 74° 50' 15° 20' 9.4223 9.9843 9.4381 0.5619 74° 40' 15° 30' 9.4269 9.9839 9.4430 0.5570 74° 30' 15° 40' 9.4314 9.9836 9.4479 0.5521 74° 20' 15° 50' 9.4359 9.9832 9.4527 0.5473 74° 10' 16° 00' 9.4403 9.9828 9.4575 0.5425 74° 00' 16° 10' 9 .4447 9.9825 9.4622 0.5378 73° 50' 16° 20' 9.4491 9.9821 9.4669 0.5331 73° 40' 16° 30' 9.4533 9.9817 9.4716 0.5284 73° 30' 16° 40' 9.4576 9.9814 9.4762 0.5238 73° 20' 16° 50' 9.4618 9.9810 9.4808 0.5192 73° 10' cos sin cot tan Degrees 200 ENGINEERING MATHEMATICS LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 17° 00' 9.4659 9.9806 9.4853 0.5147 73° 00' 17° 10' 9.4700 9.9802 9.4898 0.5102 72° 50' 17° 20' 9.4741 9.9798 9.4943 0.5057 72° 40' 17° 30' 9.4781 9.9794 9.4987 0.5013 72° 30' 17° 40' 9.4821 9.9790 9.5031 0.4969 72° 20' 17° 50' 9.4861 9.9786 9.5075 0.4925 72° 10' 18° 00' 9.4900 9.9782 9.5118 0.4882 72° 00' 18° 10' 9.4939 9.9778 9.5161 0.4839 71° 50' 18° 20' 9.4977 9.9774 9.5203 0.4797 71° 40' 18° 30' 9.5015 9.9770 9.5245 0.4755 71° 30' 18° 40' 9.5052 9.9765 9.5287 0.4713 71° 20' 18° 50' 9.5090 9.9761 9.5329 0.4671 71° 10' 19° 00' 9.5126 9.9757 9.5370 0.4630 71° 00' 19° 10' 9.5163 9.9752 9.5411 0.4589 70° 50' 19° 20' 9.5199 9.9748 9.5451 0.4549 70° 40' 19° 30' 9.5235 9.9743 9.5491 0.4509 70° 30' 19° 40' 9.5270 9.9739 9.5531 0.4469 70° 20' 19° 50' 9.5306 9.9734 • 9.5571 0.4429 70° 10' 20° 00' 9.5341 9.9730 9.5611 0.4389 70° 00' 20° 10' 9.5375 9.9725 9.5650 0.4350 69° 50' 20° 20' 9.5409 9.9721 9.5689 0.4311 69° 40' 20° 30' 9.5443 9.9716 9.5727 0.4273 69° 30' 20° 40' 9.5477 9.9711 9.5766 0.4234 69° 20' 20° 50' 9.5510 9.9706 9.5804 0.4196 69° 10' 21° 00' • 9.5543 9.9702 9.5842 0.4158 69° 00' 21° 10' 9.5576 9.9697 9.5879 0.4121 68° 50' 21° 20' 9.5609 9.9692 9.5917 0.4083 68° 40' 21° 30' 9.5641 9.9687 9.5954 0.4046 68° 30' 21° 40' 9.5673 9.9682 9.5991 0.4009 68° 20' 21° 50' 9.5704 9.9677 9.6028 0.3972 68° 10' 22° 00' 9.5736 9.9672 9.6064 0.3936 68° 00' 22° 10' 9.5767 9.9667 9.6100 0.3900 67° 50' 22° 20' 9.5798 9.9661 9.6136 0.3864 67° 40' 22° 30' 9.5828 9.9656 • 9.6172 0.3828 67° 30' cos sin cot tan Degrees TABLES 201 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 22° 40' 9.5859 9.9651 9.6208 0.3792 67° 20' 22° 50' 9.5889 9.9646 9.6243 0.3757 67° 10' 23° 00' 9.5919 9.9640 9.6279 0.3721 67° 00' 23° 10' 9.5948 9.9635 9.6314 0.3686 66° 50' 23° 20' 9.5978 9.9629 9.6348 0.3652 66° 40' 23° 30' 9.6007 9.9624 9.6383 0.3617 66° 30' 23° 40' 9.6036 9.9618 9.6417 0.3583 66° 20' 23° 50' 9.6065 9.9613 9.6452 0.3548 66° 10' 24° 00' 9.6093 9.9607 9.6486 0.3514 66° 00' 24° 10' 9.6121 9.9602 9.6520 0.3480 65° 50' 24° 20' 9.6149 9.9596 9.6553 0.3447 65° 40' 24° 30' 9.6177 9.9590 9.6587 0.3413 65° 30' 24° 40' 9.6205 9.9584 9.6620 0.3380 65° 20' 24° 50' 9.6232 9.9579 9.6654 0.3346 65° 10' 25° 00' 9.6259 9.9573 9.6687 0.3313 65° 00' 25° 10' 9.6286 9.9567 9.6720 0.3280 64° 50' 25° 20' 9.6313 9.9561 9.6752 0.3248 64° 40' 25° 30' 9.6340 9.9555 9.6785 0.3215 64° 30' 25° 40' 9.6366 9.9549 9.6817 0.3183 64° 20' 25° 50' 9.6392 9.9543 9.6850 0.3150 64° 10' 26° 00' 9.6418 9.9537 9.6882 0.3118 64° 00' 26° 10' 9.6444 9.9530 9.6914 0.3086 63° 50' 26° 20' 9.6470 9.9524 9.6946 0.3054 63° 40' 26° 30' 9.6495 9.9518 9.6977 0.3023 63° 30' 26° 40' 9.6521 9.9512 9.7009 0.2991 63° 20' 26° 50' 9.6546 9.9505 9.7040 0.2960 63° 10' 27° 00' 9.6570 9.9499 9.7072 0.2928 63° 00' 27° 10' 9.6595 9.9492 9.7103 0.2897 62° 50' 27° 20' 9.6620 9.9486 9.7134 0.2866 62° 40' 27° 30' 9.6644 9.9479 9.7165 0.2835 62° 30' 27° 40' 9.6668 9.9473 9.7196 0.2804 62° 20' 27° 50' 9.6692 9.9466 9.7226 0.2774 62° 10' 28° 00' 9.6716 9.9459 9.7257 0.2743 62° 00' 28° 10' 9.6740 9.9453 9.7287 0.2713 61° 50' cos sin cot tan Degrees 202 ENGINEERING MATHEMATICS LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS {:Continued) Degrees sin COS tan cot 28° 20' 9.6763 9.9446 9.7317 0.2683 61° 40' 28° 30' 9.6787 9.9439 9.7348 0.2652 61° 30' 28° 40' 9.6810 9.9432 9.7378 0.2622 61° 20' 28° 50' 9.6833 9.9425 9.7408 0.2592 61° 10' 29° 00' 9.6856 9.9418 9.7438 0.2562 61° 00' 29° 10' 9.6878 9.9411 9.7467 0.2533 60° 50' 29° 20' 9.6901 9.9404 9.7497 0.2503 60° 40' 29° 30' 9.6923 9.9397 9.7526 0.2474 60° 30' 29° 40' 9.6946 9.9390 9.7556 0.2444 60° 20' 29° 50' 9.6968 9.9383 9.7585 0.2415 60° 10' 30° 00' 9.6990 9.9375 9.7614 0.2386 60° 00' 30° 10' 9.7012 9.9368 9.7644 0.2356 59° 50' 30° 20' 9.7033 9.9361 9.7673 0.2327 59° 40' 30° 30' 9.7055 9.9353 9.7701 0.2299 59° 30' 30° 40' 9.7076 9.9346 9.7730 0.2270 59° 20' 30° 50' 9.7097 9.9338 9.7759 0.2241 59° 10' 31° 00' 9.7118 9.9331 9.7788 0.2212 59° 00' 31° 10' 9.7139 9.9323 9.7816 0.2184 58° 50' 31° 20' 9.7160 9.9315 9.7845 0.2155 58° 40' 31° 30' 9.7181 9.9308 9.7873 0.2127 58° 30' 31° 40' 9.7201 9.9300 9.7902 0.2098 58° 20' 31° 50' 9.7222 9.9292 9.7930 0.2070 58° 10' 32° 00' 9.7242 9.9284 9.7958 0.2042 58° 00' 32° 10' 9.7262 9.9276 9.7986 0.2014 57° 50' 32° 20' 9.7282 9.9268 9.8014 0.1986 57° 40' 32° 30' 9.7302 9.9260 9.8042 0.1958 57° 30' 32° 40' 9.7322 9.9252 9.8070 0.1930 57° 20' 32° 50' 9.7342 9.9244 9.8097 0.1903 57° 10' 33° 00' 9.7361 9.9236 9.8125 0.1875 57° 00' 33° 10' 9.7380 9.9228 9.8153 0.1847 56° 50' 33° 20' 9.7400 9.9219 9.8180 0.1820 56° 40' 33° 30' 9.7419 9.9211 9.8208 0.1792 56° 30' 33° 40' 9.7438 9.9203 9.8235 0.1765 56° 20' 33° 50' 9.7457 9.9194 9.8263 0.1737 56° 10' cos sin cot tan Degrees TABLES 203 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS (Continued) Degrees sin COS tan cot 34° 00' 9.7476 9.9186 9.8290 0.1710 56° 00' 34° 10' 9.7494 9.9177 9.8317 0.1683 55° 50' 34° 20' 9.7513 9.9169 9.8344 0.1656 55° 40' 34° 30' 9.7531 9.9160 9.8371 0.1629 55° 30' 34° 40' 9.7550 9.9151 9.8398 0.1602 55° 20' 34° 50' 9.7568 9.9142 9.8425 0.1575 55° 10' 35° 00' 9.7586 9.9134 9.8452 0.1548 55° 00' 35° 10' 9.7604 9.9125 9.8479 0.1521 54° 50' 35° 20' 9.7622 9.9116 9.8506 0.1494 54° 40' 35° 30' 9.7640 9.9107 9.8533 0.1467 54° 30' 35° 40' 9.7657 9.9098 9.8559 0.1441 54° 20' 35° 50' 9.7675 9.9089 9.8586 0.1414 54° 10' 36° 00' 9.7692 9.9080 9.8613 0.1387 54° 00' 36° 10' 9.7710 9.9070 9.8639 0.1361 53° 50' 36° 20' 9.7727 9.9061 9.8666 0.1334 53° 40' 36° 30' 9.7744 9.9052 9.8692 0.1308 53° 30' 36° 40' 9.7761 9.9042 9.8718 0.1282 53° 20' 36° 50' 9.7778 9.9033 9.8745 0.1255 53° 10' 37° 00' 9.7795 9.9023 9.8771 0.1229 53° 00' 37° 10' 9.7811 9.9014 9.8797 0.1203 52° 50' 37° 20' 9.7828 9.9004 9.8824 0.1176 52° 40' 37° 30' 9.7844 9.8995 9.8850 0.1150 52° 30' 37° 40' 9.7861 9.8985 9.8876 0.1124 52° 20' 37° 50' 9.7877 9.8975 9.8902 0.1098 52° 10' 38° 00' 9.7893 9.8965 9.8928 0.1072 52° 00' 38° 10' 9.7910 9.8955 9.8954 0.1046 51° 50' 38° 20' 9.7926 9.8945 9.8980 0.1020 51° 40' 38° 30' 9.7941 9.8935 9.9006 0.0994 51° 30' 38° 40' 9.7957 9.8925 9.9032 0.0968 51° 20' 38° 50' 9.7973 9.8915 9.9058 0.0942 51° 10' 39° 00' 9.7989 9.8905 9.9084 0.0916 51° 00' 39° 10' 9.8004 9.8895 9.9110 0.0890 50° 50' 39° 20' 9.8020 9.8884 9.9135 0.0865 50° 40' 39° 30^ 9.8035 9.8874 9.9161 0.0839 50° 30' 39° 40' 9.8050 9.8864 9.9187 0.0813 50° 20' 39° 50' 9.8066 9.8853 9.9212 0.0788 50° 10' cos sin cot tan Degrees 204 ENGINEERING MATHEMATICS LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS (Continued) Degrees sin COS tan cot 40° 00' 9.8081 9.8843 9.9238 0.0762 50° 00' 40° 10' 9.8096 9.8832 9.9264 0.0736 49° 50' 40° 20' 9.8111 9.8821 9.9289 0.0711 49° 40' 40° 30' 9.8125 9.8810 9.9315 0.0685 49° 30' 40° 40' 9.8140 9.8800 9.9341 0.0659 49° 20' 40° 50' 9.8155 9.8789 9.9366 0.0634 49° 10' 41° 00' 9.8169 9.8778 9.9392 0.0608 49° 00' 41° 10' 9.8184 9.8767 9.9417 0.0583 48° 50' 41° 20' 9.8198 9.8756 9.9443 0.0557 48° 40' 41° 30' 9.8213 9.8745 9.9468 0.0532 48° 30' 41° 40' 9.8227 9.8733 9.9494 0.0506 48° 20' 41° 50' 9.8241 9.8722 9.9519 0.0481 48° 10' 42° 00' 9.8255 9.8711 9.9544 0.0456 48° 00' 42° 10' 9.8269 9.8699 9.9570 0.0430 47° 50' 42° 20' 9.8283 9.8688 9.9595 0.0405 47° 40' 42° 30' 9.8297 9.8676 9.9621 0.0379 47° 30' 42° 40' 9.8311 9.8665 9.9646 0.0354 47° 20' 42° 50' 9.8324 9.8653 9.9671 0.0329 47° 10' 43° 00' 9.8338 9.8641 9.9697 0.0303 47° 00' 43° 10' 9.8351 9.8629 9.9722 0.0278 46° 50' 43° 20' 9.8365 9.8618 9.9747 0.0253 46° 40' 43° 30' 9.8378 9.8606 9.9772 0.0228 46° 30' 43° 40' 9.8391 9.8594 9.9798 0.0202 46° 20' 43° 50' 9.8405 9.8582 9.9823 0.0177 46° 10' 44° 00' 9.8418 9.8569 9.9848 0.0152 46° 00' 44° 10' 9.8431 9.8557 9.9874 0.0126 45° 50' 44° 20' 9.8444 9.8545 9.9899 0.0101 45° 40' 44° 30' 9.8457 9.8532 9.9924 0.0076 45° 30' 44° 40' 9.8469 9.8520 9.9949 0.0051 45° 20' 44° 50' 9.8482 9.8507 9.9975 0.0025 45° 10' 45° 00' 9.8495 9.8495 0.0000 0.0000 45° 00' cos sin cot tan Degrees TABLES 205 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS Degrees sin COS tan cot 0°00' .0000 1.0000 .0000 00 90° 00' 0°10' .0029 1.0000 .0029 343.77 89° 50' 0°20' .0058 1.0000 .0058 171.89 89° 40' 0°30' .0087 1.0000 .0087 114.59 89° 30' 0°40' .0116 .9999 .0116 85.940 89° 20' 0°50' .0145 .9999 .0145 68.750 89° 10' 1°00' .0175 .9998 .0175 57.290 89° 00' 1° 10' .0204 .9998 .0204 49.104 88° 50' 1°20' .0233 .9997 .0233 42.964 88° 40' 1°30' .0262 .9997 .0262 38.188 88° 30' 1°40' .0291 .9996 .0291 34.368 88° 20' 1°50' .0320 .9995 .0320 31.242 88° 10' 2° 00' .0349 .9994 .0349 28.636 88° 00' 2° 10' .0378 .9993 .0378 26.432 87° 50' 2° 20' .0407 .9992 .0407 24.542 87° 40' 2° 30' .0436 .9990 .0437 22 .904 87° 30' 2° 40' .0465 .9989 .0466 21.470 87° 20' 2° 50' .0494 .9988 .0495 20.206 87° 10' 3° 00' .0523 .9986 .0524 19.081 87° 00' 3° 10' .0552 .9985 .0553 18.075 86° 50' 3° 20' .0581 .9983 .0582 17.169 86° 40' 3° 30' .0610 .9981 .0612 16.350 86° 30' 3° 40' .0640 .9980 .0641 15.605 86° 20' 3° 50' .0669 .9978 .0670 14.924 86° 10' 4° 00' .0698 .9976 .0699 14.301 86° 00' 4° 10' .0727 .9974 .0729 13.727 85° 50' 4° 20' .0756 .9971 .0758 13.197 85° 40' 4° 30' .0785 .9969 .0787 12.706 85° 30' 4° 40' .0814 .9967 .0816 12.251 85° 20' 4° 50' .0843 .9964 .0846 11.826 85° 10' 5° 00' .0872 .9962 .0875 11.430 85° 00' 5° 10' .0901 .9959 .0904 11.059 84° 50' 5° 20' .0929 .9957 .0934 10.712 84° 40' 5° 30' .0958 .9954 .0963 10.385 84° 30' cos sin cot tan Degrees 2o6 ENGINEERING MATHEMATICS NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 5° 40' .0987 .9951 .0992 10.078 84° 20' 5° 50' .1016 .9948 .1022 9.7882 84° 10' 6° 00' .1045 .9945 .1051 9.5144 84° 00' 6° 10' .1074 .9942 .1080 9.2553 83° 50' 6° 20' .1103 .9939 .1110 9.0098 83° 40' 6° 30' .1132 .9936 .1139 8.7769 83° 30' 6° 40' .1161 .9932 .1169 8.5555 83° 20' 6° 50' .1190 .9929 .1198 8.3450 83° 10' 7° 00' .1219 .9925 .1228 8.1443 83° 00' 7° 10' .1248 .9922 .1257 7.9530 82° 50' 7° 20' .1276 .9918 .1287 7.7704 82° 40' 7° 30' .1305 .9914 .1317 7.5958 82° 30' 7° 40' .1334 .9911 .1346 7.4287 82° 20' 7° 50' .1363 .9907 .1376 7.2687 82° 10' 8° 00' .1392 .9903 .1405 7.1154 82° 00' 8° 10' .1421 .9899 .1435 6.9682 81^50' 8° 20' .1449 .9894 .1465 6.8269 81° 40' 8° 30' .1478 .9890 .1495 6.6912 81° 30' 8° 40' .1507 .9886 .1524 6.5606 81° 20' 8° 50' .1536 .9881 .1554 6.4348 81° 10' 9° 00' .1564 .9877 .1584 6.3138 81° 00' 9° 10' .1593 .9872 .1614 6.1970 80° 50' 9° 20' .1622 .9868 .1644 6.0844 80° 40' 9° 30' .1650 .9863 .1673 5.9758 80° 30' 9° 40' .1679 .9858 .1703 5.8708 80° 20' 9° 50' .1708 .9853 .1733 5.7694 80° 10' 10° 00' .1736 .9848 .1763 5.6713 80° 00' 10° 10' .1765 .9843 .1793 5.5764 79° 50' 10° 20' .1794 .9838 .1823 5.4845 79° 40' 10° 30' .1822 .9833 .1853 5.3955 79° 30' 10° 40' .1851 .9827 .1883 5.3093 79° 20' 10° 50' .1880 .9822 .1914 5.2257 79° 10' 11° 00' .1908 .9816 .1944 5.1446 79° 00' 11° 10' .1937 .9811 .1974 5.0658 78° 50' cos sin cot tan Degrees TABLES 207 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 11° 20' .1965 .9805 .2004 4.9894 78° 40' 11° 30' .1994 .9799 .2035 4.9152 78° 30' 11° 40' .2022 .9793 .2065 4.8430 78° 20' 11° 50' .2051 .9787 .2095 4.7729 78° 10' 12° 00' .2079 .9781 .2126 4.7046 78° 00' 12° 10' .2108 .9775 .2156 4.6382 77° 50' 12° 20' .2136 .9769 .2186 4.5736 77° 40' 12° 30' .2164 .9763 .2217 4.5107 77° 30' 12° 40' .2193 .9757 .2247 4.4494 77° 20' 12° 50' .2221 .9750 .2278 4.3897 77° 10' 13° 00' .2250 .9744 .2309 4.3315 77° 00' 13° 10' .2278 .9737 .2339 4.2747 76° 50' 13° 20' .2306 .9730 .2370 4.2193 76° 40' 13° 30' .2334 .9724 .2401 4.1653 76° 30' 13° 40' .2363 .9717 .2432 4.1126 76° 20' 13° 50' .2391 .9710 .2462 4.0611 76° 10' 14° 00' .2419 .9703 .2493 4.0108 76° 00' 14° 10' .2447 .9696 .2524 3.9617 75° 50' 14° 20' .2476 .9689 .2555 3.9136 75° 40' 14° 30' .2504 .9681 .2586 3.8667 75° 30' 14° 40' .2532 .9674 .2617 3.8208 75° 20' 14° 50' .2560 .9667 .2648 3.7760 75° 10' 15° 00' .2588 .9659 .2679 3.7321 75° 00' 15° 10' .2616 .9652 .2711 3.6891 74° 50' 15° 20' .2644 .9644 .2742 3.6470 74° 40' 15° 30' .2672 .9636 .2773 3.6059 74° 30' 15° 40' .2700 .9628 .2805 3.5656 74° 20' 15° 50' .2728 .9621 .2836 3.5261 74° 10' 16° 00' .2756 .9613 .2867 3.4874 74° 00' 16° 10' .2784 .9605 .2899 3.4495 73° 50' 16° 20' .2812 .9596 .2931 3.4124 73° 40' 16° 30' .2840 .9588 ,2962 3.3759 73° 30' 16° 40' .2868 .9580 .2994 3.3402 73° 20' 16° 50' .2896 .9572 .3026 3.3052 73° 10' cos sin cot tan Degrees 208 ENGINEERING MATHEMATICS NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS (Continued) Degrees sin COS tan cot 17° 00' .2924 .9563 .3057 3.2709 73° 00' 17° 10' .2952 .9555 .3089 3.2371 72° 50' 17° 20' .2979 .9546 .3121 3.2041 72° 40' 17° 30' .3007 .9537 .3153 3.1716 72° 30' 17° 40' .3035 .9528 .3185 3 . 1397 72° 20' 17° 50' .3062 .9520 .3217 3.1084 72° 10' 18° 00' .3090 .9511 .3249 3.0777 72° 00' 18° 10' .3118 .9502 .3281 3.0475 71° 50' 18° 20' .3145 .9492 .3314 3.0178 71° 40' 18° 30' .3173 .9483 .3346 2.9887 71° 30' 18° 40' .3201 .9474 .3378 2.9600 71° 20' 18° 50' .3228 .9465 .3411 2.9319 71° 10' 19° 00' .3256 .9455 .3443 2.9042 71° 00' 19° 10' .3283 .9446 .3476 2.8770 70° 50' 19° 20' .3311 .9436 .3508 2.8502 70° 40' 19° 30' .3338 .9426 .3541 2.8239 70° 30' 19° 40' .3365 .9417 .3574 2.7980 70° 20' 19° 50' .3393 .9407 .3607 2.7725 70° 10' 20° 00' .3420 .9397 .3640 2.7475 70° 00' 20° 10' .3448 .9387 # .3673 2.7228 69° 50' 20° 20' .3475 .9377 .3706 2.6985 69° 40' 20° 30' .3502 .9367 .3739 2.6746 69° 30' 20° 40' .3529 .9356 .3772 2.6511 69° 20' 20° 50' .3557 .9346 .3805 2.6279 69° 10' 21° 00' .3584 .9336 .3839 2.6051 69° 00' 21° 10' .3611 .9325 .3872 2.5826 68° 50' 21° 20' .3638 .9315 .3906 2.5605 68° 40' 21° 30' .3665 .9304 .3939 2.5386 68° 30' 21° 40' .3692 .9293 .3973 2.5172 68° 20' 21° 50' .3719 .9283 .4006 2.4960 68° 10' 22° 00' .3746 .9272 .4040 2.4751 68° 00' 22° 10' .3773 .9261 .4074 2.4545 67° 50' 22° 20' .3800 .9250 .4108 2.4342 67° 40' 22° 30' .3827 .9239 .4142 2.4142 67° 30' cos sin cot tan Degrees TABLES 209 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 22° 40' .3854 .9228 .4176 2.3945 67° 20' 22° 50' .3881 .9216 .4210 2.3750 67° 10' 23° 00' .3907 .9205 .4245 2.3559 67° 00' 23° 10' .3934 .9194 .4279 2.3369 66° 50' 23° 20' .3961 .9182 .4314 2.3183 66° 40' 23° 30' .3987 .9171 .4348 2.2998 66° 30' 23° 40' .4014 .9159 .4383 2.2817 66° 20' 23° 50' .4041 .9147 .4417 2.2637 66° 10' 24° 00' .4067 .9135 .4452 2.2460 66° 00' 24° 10' .4094 .9124 .4487 2.2286 65° 50' 24° 20' .4120 .9112 .4522 2.2113 65° 40' 24° 30' .4147 .9100 .4557 2.1943 65° 30' 24° 40' .4173 .9088 .4592 2.1775 65° 20' 24° 50' .4200 .9075 .4628 2.1609 65° 10' 25° 00' .4226 .9063 .4663 2.1445 65° 00' 25° 10' .4253 .9051 .4699 2 . 1283 64° 50' 25° 20' .4279 .9038 .4734 2.1123 64° 40' 25° 30' .4305 .9026 .4770 2.0965 64° 30' 25° 40' .4331 .9013 .4806 2.0809 64° 20' 25° 50' .4358 .9001 .4841 2.0655 64° 10' 26° 00' .4384 .8988 .4877 2.0503 64° 00' 26° 10' .4410 .8975 .4913 2.0353 63° 50' 26° 20' .4436 .8962 .4950 2.0204 63° 40' 26° 30' .4462 .8949 .4986 2.0057 63° 30' 26° 40' .4488 .8936 .5022 1.9912 63° 20' 26° 50' .4514 .8923 .5059 1.9768 63° 10' 27° 00' .4540 .8910 .5095 1.9626 63° 00' 27° 10' .4566 .8897 .5132 1.9486 62° 50' 27° 20' .4592 .8884 .5169 1.9347 62° 40' 27° 30' .4617 .8870 .5206 1.9210 62° 30' 27° 40' .4643 .8857 .5243 1.9074 62° 20' 27° 50' .4669 .8843 .5280 1.8940 62° 10' 28° 00' .4695 .8829 .5317 1.8807 62° 00' 28° 10' .4720 .8816 .5354 1.8676 61° 50' cos sin cot tan Degrees 2IO ENGINEERING MATHEMATICS NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 28° 20' .4746 .8802 .5392 1.8546 61° 40' 28° 30' .4772 .8788 .5430 1.8418 61° 30' 28° 40' .4797 .8774 .5467 1.8291 61° 20' 28° 50' .4823 .8760 .5505 1.8165 61° 10' 29° 00' .4848 .8746 .5543 1.8040 61° 00' 29° 10'. .4874 .8732 .5581 1.7917 60° 50' 29° 20' .4899 .8718 .5619 1.7796 60° 40' 29° 30' .4924 .8704 .5658 1.7675 60° 30' 29° 40' .4950 .8689 .5696 1.7556 60° 20' 29° 50' .4975 .8675 .5735 1.7437 60° 10' 30° 00' .5000 .8660 .5774 1.7321 60° 00' 30° 10' .5025 .8646 .5812 1.7205 59° 50' 30° 20' .5050 .8631 .5851 1.7090 59° 40' 30° 30' .5075 .8616 .5890 1.6977 59° 30' 30° 40' .5100 .8601 .5930 1.6864 59° 20' 30° 50' .5125 .8587 .5969 1.6753 59° 10' 31° 00' .5150 .8572 .6009 1.6643 59° 00' 31° 10' .5175 .8557 .6048 1.6534 58° 50' 31° 20' .5200 .8542 .6088 1.6426 58° 40' 31° 30' .5225 .8526 .6128 1.6319 58° 30' 31° 40' .5250 8511 .6168 1.6212 58° 20' 31° 50' .5275 .8496 .6208 1.6107 58° 10' 32° 00' .5299 .8480 .6249 1.6003 58° 00' 32° 10' .5324 .8465 .6289 1.5900 57° 50' 32° 20' .5348 .8450 .6330 1.5798 57° 40' 32° 30' .5373 .8434 .6371 1.5697 57° 30' 32° 40' .5398 .8418 .6412 1.5597 57° 20' 32° 50' .5422 .8403 .6453 1.5497 57° 10' 33° 00' .5446 .8387 .6494 1.5399 57° 00' 33° 10' .5471 .8371 .6536 1.5301 56° 50' 33° 20' .5495 .8355 .6577 1.5204 56° 40' 33° 30' .5519 .8339 .6619 1.5108 56° 30' 33° 40' .5544 .8323 .6661 1.5013 56° 20' 33° 50' .5568 .8307 .6703 1.4919 56° 10' cos sin cot tan Degrees TABLES 211 NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 34° 00' .5592 .8290 .6745 1.4826 56° 00' 34° 10' .5616 .8274 .6787 1.4733 55° 50' 34° 20' .5640 .8258 .6830 1.4641 55° 40' 34° 30' .5664 .8241 .6873 1.4550 55° 30' 34° 40' .5688 .8225 .6916 1.4460 55° 20' 34° 50' .5712 .8208 .6959 1.4370 55° 10' 35° 00' .5736 .8192 .7002 1.4281 55° 00' 35° 10' .5760 .8175 .7046 1 .4193 54° 50' 35° 20' .5783 .8158 .7089 1.4106 54° 40' 35° 30' .5807 .8141 .7133 1.4019 54° 30' 35° 40' .5831 .8124 .7177 1.3934 54° 20' 35° 50' .5854 .8107 .7221 1.3848 54° 10' 36° 00' .5878 .8090 .7265 1.3764 54° 00' 36° 10' .5901 .8073 .7310 1.3680 53° 50' 36° 20' .5925 .8056 .7355 1.3597 53° 40' 36° 30' .5948 .8039 .7400 1.3514 53° 30' 36° 40' .5972 .8021 .7445 1.3432 53° 20' 36° 50' .5995 .8004 .7490 1.3351 53° 10' 37° 00' .6018 .7986 .7536 1.3270 53° 00' 37° 10' .6041 .7969 .7581 1.3190 52° 50' 37° 20' .6065 .7951 .7627 1.3111 52° 40' 37° 30' .6088 .7934 .7673 1.3032 52° 30' 37° 40' .6111 .7916 .7720 1.2954 52° 20' 37° 50' .6134 .7898 .7766 1.2876 52° 10' 38° 00' .6157 .7880 .7813 1.2799 52° 00' 38° 10' .6180 .7862 .7860 1.2723 51° 50' 38° 20' .6202 .7844 .7907 1.2647 51° 40' ?>%'' 30' .6225 .7826 .7954 1.2572 51° 30' 38° 40' .6248 .7808 .8002 1.2497 51° 20' 38° 50' .6271 .7790 .8050 1.2423 51° 10' 39° 00' .6293 .7771 .8098 1.2349 51° 00' 39° 10' .6316 .7753 .8146 1.2276 50° 50' 39° 20' .6338 .7735 .8195 1.2203 50° 40' 39° 30' .6361 .7716 .8243 1.2131 50° 30' cos sin cot tan Degrees 212 ENGINEERING MATHEMATICS NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS {Continued) Degrees sin COS tan cot 39° 40' .6383 .7698 .8292 1.2059 50° 20' 39° 50' .6406 .7679 .8342 1 . 1988 50° 10' 40° 00' .6428 .7660 .8391 1.1918 50° 00' 40° 10' .6450 .7642 .8441 1.1847 49° 50' 40° 20' .6472 .7623 .8491 1.1778 49° 40' 40° 30' .6494 .7604 .8541 1.1708 49° 30' 40° 40' .6517 .7585 .8591 1 . 1640 49° 20' 40° 50' .6539 .7566 .8642 1.1571 49° 10' ^ 41° 00' .6561 .7547 .8693 1.1504 49° 00' 41° 10' .6583 .7528 .8744 1.1436 48° 50' 41° 20' .6604 .7509 .8796 1.1369 48° 40' 41° 30' .6626 .7490 .8847 1.1303 48° 30' 41° 40' .6648 .7470 .8899 1.1237 48° 20' 41° 50' .6670 .7451 .8952 1.1171 ^48° 10' 42° 00' .6691 .7431 .9004 1.1106 48° 00' 42° 10' .6713 .7412 .9057 1.1041 47° 50' 42° 20' .6734 .7392 .9110 1.0977 47° 40' 42° 30' .6756 .7373 .9163 1.0913 47° 30' 42° 40' .6777 .7353 .9217 1.0850 47° 20' 42° 50' .6799 .7333 .9271 1.0786 47° 10' 43° 00' .6820 .7314 .9325 1.0724 47° 00' 43° 10' .6841 .7294 .9380 1.0661 46° 50' 43° 20' .6862 .7274 .9435 1.0599 46° 40' 43° 30' .6884 .7254 .9490 1.0538 46° 30' 43° 40' .6905 .7234 .9545 1.0477 46° 20' 43° 50' .6926 .7214 .9601 1.0416 46° 10' 44° 00' .6947 .7193 .9657 1.0355 46° 00' 44° 10' .6967 .7173 .9713 1 .0295 45° 50' 44° 20' .6988 .7153 .9770 1.0235 45° 40' 44° 30' .7009 .7133 .9827 1.0176 45° 30' 44° 40' .7030 .7112 .9884 1.0117 45° 20' 44° 50' .7050 .7092 .9942 1.0058 45° 10' 45° 00' .7071 .7071 1.0000 1.0000 45° 00' cos sin cot tan Degrees TABLES 213 HYPERBOLIC SINES AND COSINES n cosh n sinh n n cosh n sinh n 0.00 1.0000 0.0000 2.05 3.9484 3.8196 0.05 1.0013 0.0500 2.10 4.1443 4.0219 0.10 1.0050 0.1002 2.15 4.3507 4.2342 0.15 1.0112 0.1506 2.20 4.5679 4.4571 0.20 1.0201 0.2013 2.25 4.7966 4.6912 0.25 1.0314 0.2526 2.30 5.0372 4.9369 0.30 1.0453 0.3045 2.35 5.2905 5 . 1952 0.35 1.0619 0.3572 2.40 5.5569 5.4662 0.40 1.0811 0.4108 2.45 5.8373 5.7510 0.45 1 . 1030 0.4653 2.50 6.1323 6.0502 0.50 1.1276 0.5211 2.55 6.4426 6.3645 0.55 1.1551 0.5782 2.60 6.7690 6.6947 0.60 1 . 1855 0.6367 2.65 7.1123 7.0417 0.65 1.2188 0.6967 2.70 7.4735 7.4063 0.70 1.2552 0.7586 2.75 7.8533 7.7894 0.75 1.2947 0.8223 2.80 8.2527 8.1919 0.80 1.3374 0.8881 2.85 8.6728 8.6150 0.85 1.3835 0.9561 2.90 9.1146 9.0596 0.90 1.4331 1.0265 2.95 9.5791 9.5268 0.95 1.4862 1.0995 3.00 10.0677 10.0179 1.00 1.5431 1.1752 3.05 10.5814 10.5340 1.05 1.6038 1.2539 3.10 11.1215 11.0765 1.10 1.6685 1.3356 3.15 11.6895 11.6466 1.15 1.7374 1.4208 3.20 12.2866 12.2459 1.20 1.8107 1.5097 3.25 12.9146 12.8758 1.25 1.8884 1.6019 3.30 13.5748 13.5379 1.30 1.9709 1.6984 3.35 14.2689 14.2338 1.35 2.0583 1.7991 3.40 14.9987 14.9654 1.40 2.1509 1.9043 3.45 15.7661 15.7343 1.45 2.2488 2.0143 3.50 16.5728 16.5426 1.50 2.3524 2.1293 3.55 17.4210 17.3923 1.55 2.4619 2.2496 3.60 18.3128 18.2855 1.60 2.5775 2.3757 3.65 19.2503 19.2243 1.65 2.6995 2.5075 3.70 20.2360 20.2113 1.70 2.8283 2.6456 3.75 21.2723 21.2488 1.75 2.9642 2.7904 3.80 22.3618 22.3394 1.80 3.1075 2.9422 3.85 23.5072 23.4859 1.85 3.2583 3.1013 3.90 24.7113 24.6911 1.90 3.4177 3.2682 3.95 25.9773 25.9581 1.95 3.5855 3.4432 4.00 27.3082 27.2899 2.00 3.7622 3.6269 214 ENGINEERING MATHEMATICS Numerical Constants TT = 3.141 592 654 logioir = 0.497 149 873 = 0.318 309 886 1 ■K t" = 9.869 604 401 VV = 1.772 453 851 e = 2.718 281828 logio e = 0.434 294 482 loge 10 = 2.302 585 093 logiologioe = 9.637 784 311 logeTT = 1.144 729 886 loge 2 = 0.693 147 181 logio 2 = 0.301 029 996 INDEX Acceleration, 78, 79, 80. Admittance, electric, 151. Air: flow through apertures, 131. flow through pipes, 132, 133. Algebra, 1-9. Alternating currents, 147-154. Altitude, determination of, 167. Ampere, 165. Analytic geometry, 19-30. Angle between two lines, 29. Angular distortion, 94. Angular measure, 82. Apertures, flow through: air and steam, 131-132. water, 120-12 1. Apothecaries' measure, 159. Arc, length of, 31, 32, 36. Areas: integral formulae for, 37, 38. of circles, table of, 185-186. Arithmetical progression, 2. Asymptotes of hyperbola, 25. Atmosphere, standard, 165. Atomic weights, 168. Avoirdupois weight, 159. Barometer: reduction of readings to 0° C, 166. determination of altitudes by, 167. Bazin's formulae for flow of water: in channels, 127-128. over weirs, 129-130. Beam loadings, 98-106. Beams: cantilever, 99. deflection of, 97. flexure of, 95-97. Belt friction, 89. Bending moment of a beam, 96. Bernoulli's theorem, hydraulics, 121-122. Binomial theorem, 2. Boiling points of elements, 172- 173. British thermal unit, 164, 174. Calculus, 30-54. Calorie, 164. Cantilever beam, 99. Capacity: electric, 145-147, 150. measures of, 159, 163. Catenary, 27. Center of gravity: composite sections, 74, 75. formulae for, 72-74. standard sections, table of, iii- 113. Center of gyration, 77. Center of percussion, 77. Center of pressure, 119, 120. Centigrade thermometer, 163. Centrifugal force, 83. Channel beam, 117. Channels, flow of water in, 127- 129. Characteristic of a logarithm, 190. Chezy's formula, flow of water, 127. Chord of circle, 11. Circle: circumference and area of, 11. chord of, II. equations of, 21, 22. moment of inertia of, 116. sector and segment of, 11. tangent to, 21. Circles, table of circumferences and areas of, 185, 186. Circular measure, 159. Circumferences of circles, table of, 185, 186. 215 2l6 INDEX Coefl5cients of linear expansion, 175. 176. Collapsing of tubes, 131. Columns, formulae for, 1 06-1 11. Combinations and permutations, 9. Common logarithms: base of, 3. of numbers, table of, 192, 193. Complex imaginary quantities, 7. 8. Composition of forces, 87, 88. Compression: strength of materials, 93. stress due to, 92. Condensers, electric, 146, 147. Conductance, electric, 151. Cone, right circular: center of gravity of, 114. frustum of, 13. lateral surface and volume, 12. Constants, numerical, 214. Convergent series, 5. Conversion factors, English and Sietric, 160-163. Ccipper wire tables, 136, 137. Coulomb, 145. Cubes and cube roots of numbers, table of, 187-189. Cubic equations, 4, 39, 40. Curvature, radius of, 32. Curve tracing, 40, 41. Cycloid, 25. Cylinder, right circular: lateral surface and volume, 12. moment of inertia of, 115. Cylinders, stresses in, 130. Deflection of beams, 97. De Moivre's theorem, 9. Density of various substances, 169-171. Differential calculus: application to geometry and physics, 30-32. formulae, 33. Differential equations, 58-71. exact equations, 69, 70. first order and first degree, 59- 62. Differential equations — cont. first order and higher degree. 62, 63. linear equations, 63-69. second order and first degree, 70, 71. Direction cosines of a line, 27, 28, 29. Discriminant of quadratic equa- tion, I. Distance: between two points, 20, 28; 29 from point to line, 20. from point to plane, 29. Distortion, angular, stress due to, 94. Diverging series, 5, 6. Dry measure, 159. Dyne, 163. e, base of natural logarithms, 3, 6. Eddy current loss, 156 Elastic curve of a beam, 96, 97. Elasticity, modulus of, 92, 93, 94. Elastic limit, 91, 93. Electricity, 134-157. Ellipse: circumference and area of, 11. definition of, 23. equation of, 23. tangent to, 24. Energy: electric, 139. kinetic, 85, 86. of flywheel, 83, 84. EngUsh and metric conversion factors, 160-163. English weights and measures, 158, 159. Epicycloid, 26. Equations: cubic and higher, 4, 39, 40. quadratic, i. transcendental, 4, 5. Equivalents: heat, electric, and mechanical, 164-166. pressure, 165, 166 Erg, 164. Euler's formula for columns, 106, 107. INDEX 217 Exponential transformations, 9. Exponents, i. Factors of safety, table of, 93. Fahrenheit thermometer, 163. Falling bodies, 80. Farad, 145. Flexure of beams, 95-97. Fliegner'sequations,flowofair,i3i. Flow of air, 131, 132, 133. Flow of steam, 132, 133. Flow of water, 120-130. Fluids, flow of, 131-133. Flywheel, 83, 84. Force, 85, 86. Force, centrifugal, 83. Forces: composition of, 87, 88. resolution of, 88. Francis' formula, flow of water, 129. Frequency, electric dynamos, 139, 148. Friction, 88, 89. Frustum of pyramid, 12. Frustum of right circular cone, 13. g, acceleration of gravity, 80. Generators, electric, 139, 141. Geometrical progression, 2, 3. Geometry, 10-13. Gordon's formula for columns, 107, 108. Grashof's formula, flow of steam, 132. , Gravity, acceleration of, 80. Gyration, radius of, 77. Head of water, 119. Heat, electric, and mechanical equivalents, 164-166. Henry, 143. Higher degree equations, 4, 39, 40. Horsepower, i6r5. Hydraulics, 1 19-13 1. Hyperbola: asymptotes, 25. definition of, 24. equations of, 24. Hyperbolic functions, 54-58. Hyperbolic logarithms, 3, 194-196. Hyperbolic sines and cosines, table of, 213. Hypocycloid, 26, 27. Hysteresis loss, 155. I-beam, 118. Imaginary quantities, 7, 8. Impact, 86, 87. Impedance, electric, 149, 150. Impulse, 86. Inclined plane, 89-91. Indeterminate forms, i, 38, 39. Induced voltage, 142, 143. Inductance, 143-145, 149, 150. Inductivities, table of, 146. Infinite series, 5, 6, 7. Integral calculus: areas, 37, 38. length of curves, 36. volumes, 38. Integration, methods of: partial fractions, 42. parts, 43. reduction formulae, 45, 46. substitution, 43-45. table of integrals, 46-54. J. B. Johnson's formula for columns, 109, no. Joule, 164. Kinetic energy, 85, 86. Kutter's formula for flow of water, 128, 129. Length, measures of, 158, 160. Length of curves, integral formu- lae, 36. Linear expansion, coefficients of, 175, 176. Liquid measures, 159, 163. Loadings of beams, 98-106. Logarithmic cross-section paper, 189-190. Logarithms, 3, 4. Logarithms of numbers, tables of, 192-196. Logarithms of trigonometric func- tions, tables of, 197-204. 2l8 INDEX Logarithm tables, use of, 190, 191. Losses due to flow of water in pipes, 122-125. Maclaurin's series, 35. Magnetism, 154-157. Magnets, attractive force, 155. Materials, mechanics of, 91-118. Materials, strength of, table, 93. Maxima and minima, 34, 35. Measurement, tables, 158-167. Mechanics of materials, 91-118. Mechanics, theoretical, 72-91. Melting points of elements, 172, 173. Metric and English conversion factors, 160-163. Modulus of elasticity, 92, 93, 94. Mollier chart for steam, after 184. Moment of inertia: plane areas, formulae, 75, 76. solids, formula, 76. sohds, table of, 115. standard sections, table of, 116- 118. Moment of resistance of a beam, 96. Momentum, 85. Motion: circular, 82, 83. of a body, 78, 79. Motors, electric, 139, 140. Naperian logarithms, 3, 194-196. Napier's equations, flow of steam, 132. Natural logarithms: base of, 3. of numbers, tables of, 194-196. Natural trigonometric functions, 205-212. Navigation, 17-19. Neutral axis of beams, 95, 96. Neutral surface of beams, 95. Newton's method, solution of equations by, 39, 40. Normal to curve, slope of, 30. Numerical constants, 214. Oblique spherical triangles, 16, 17. Ohm, 165. Ohm's law, 138. Open channels, flow of water in, 127-129. Oscillation, radius of, 77, 78. Overfall- weirs, 129, 130. Parabola: arc and area of, 11. center of gravity of, 113. definition of, 22. equations of, 22, 23. tangent to, 23. Parallelogram, 10. Pendulum, 85. Percussion, center of, 77. Permeability, 154. Permutations and combinations, 9. Phase, alternating currents, 148, 151- Physical and chemical constants, 168-184. Pipes: flow of water in, 1 21-127. stresses in, 130. Plane analytic geometry, 19-27. Plane, equations of, 29. Plane trigonometry, 13-16. Polar coordinates, transformation of, 20. Polygon, 10. Poundal, 163. Power, electric, 139, 152, 153, 154. Powers of numbers, table of, 187- 189. Pressure: barometer readings, 166. center of, 119, 120. equivalents, 165, 166. pipes, 130. water, 119. Prism, right, 12. Progression: arithmetical, 2. geometrical, 2, 3. Projectiles, 80-82. Pyramid: center of gravity of, 114. frustum of, 12. lateral area and volume of, 12. INDEX 219 Quadratic equations, i. Radian, 82. Radius of curvature: formula for, 32. of beams, 97. Radius of gyration: formula for, 77. of standard sections, 116-118. Radius of oscillation, 77, 78. Rankine's formula for columns, 107-108. Ratio test for infinite series, 6. Reactance, electric, 149, 150. Reciprocals of numbers, table of, 187-189. Rectangular coordinates, trans- formation of, 20. Reduction formulae, integration by, 45, 46. Regular polygon, 10. Reluctance, 154. Resistance, electric: copper wire tables, 136, 137. formulae, 134, 135, 138, 139. Resistivity table, electric, 134. Resolution of forces, 88. Right circular cone, 12. Right circular cylinder: lateral surface and volume, 12. moment of inertia of, 115. Right prism, 12. Right spherical triangles, 16. Right triangle: area, 10. trigonometric formulae, 13. Ritter's formula for columns, 108, 109. Rootsof numbers, table of, 187-189. Rotation of a body, 79. 80, 82, S3. Saturated steam, tables, 177-184. Saturation curves, magnetic, 156, 157. Seconds-pendulum, 85. Section modulus of standard sec- tions, 116-118. Sector of circle: arc and area of, 11 center of gravity of, 112. Segment of circle: ar^ and chord of, 11. center of gravity of, 113. Segment of sphere, 13. Series: infinite, 5, 6, 7. Maclaurin's, 35. standard, 6, 7. Taylor's, 35. Shafts, torsion of, 94, 95. Shear, 94, 97, 98. Simple pendulum, 85. Sine wave of voltage or current, 147, 150, 151. Slope of tangent to curve, 30. Solid analytic geometry, 27-30. Solids of revolution, s^. Solution of equations, 4, 5, 39, 40. Specific heats, 173, 174. Sphere: moment of inertia of, 115. segment of, 13. surface and volume of, 13. Spherical triangles, 16, 17. Spherical trigonometry, 16-19. Squares and square roots of num- bers, table of, 187-189. Steam chart, MoUier, after 184. Steam: flow through apertures, 132. flow through pipes, 133. Steam tables, 177-184. Straight line, equations of, 19, 20, 30. Straight line formula for columns, no. Strength of materials, table of, 93. Stress, mechanical, 91, 92. Stresses in pipes and cylinders. 130, 131- Surface, measures of, 158, 160, 161. Surfaces of revolution, 37, 38. Susceptance, electric, 151. Tables, mathematical, 185-214. Tangent: to circle, 21. to ellipse, 24. to hyperbola, 25. to parabola, 23. 220 INDEX Tangent to curve, slope of, 30. Taylor's series, 35. Temperature coefficients of elec- trical resistance, table of, 135. Tension, 92. Thermometers, conversion of scales, 163. Three-phase circuits, 153, 154. Torque of motor, 141, 142. Torsion of shafts, 94, 95. Tracing curves, 40, 41. Transcendental equations, 4, 5. Transformation of coordinates, 20. Transformations, exponential- 9. Trapezoid: area of, 10. center of gravity of, 112. Triangle: area, 10. center of gravity of, iii. solution of, 16. Trigonometric formulae, 13-17. Trigonometric functions, tables of, 197-212. Trigonometry, plane, 13-16. Trigonometry, spherical, 16-19. Troy weight, 159. Tubes, collapsing of, 131. Ultimate strength, 91, 93. Units: conversion factors, 160-163. heat, electric, and mechanical equivalents, 164-166. weights and measures, 158, 159. Velocity, 78, 79- Volt, 165. Voltage, induced, 142, 143. Volume, measures of, 158, 161, 162. Volumes: reduction to standard condi- tions, 166. surfaces of revolution, 37, 38. Water, density of, 170. Watt, 165. Weight of various substances, 169- 171. Weights and measures, 158, 159. Weirs, 129, 130. 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